Worksheets/v1.1/SM_tutorial_sympy1.0.ipynb

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    "# SageMath manifold tutorial (SymPy version)\n",
    "\n",
    "This worksheet provides a short introduction to differentiable manifolds in SageMath, with [SymPy](http://www.sympy.org) as symbolic engine, instead of SageMath's default one (Pynac+Maxima). The tools described below have been implemented through the\n",
    "[SageManifolds project](http://sagemanifolds.obspm.fr) (version 1.1, included in SageMath 8.1).\n",
    "\n",
    "Click [here](https://raw.githubusercontent.com/sagemanifolds/SageManifolds/master/Worksheets/v1.1/SM_tutorial_sympy.ipynb) to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, via the command `sage -n jupyter`\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
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   "source": [
    "The following assumes that you are using version 8.1 (or higher) of SageMath, since lower versions do not include the SymPy symbolic engine for computations on manifolds:"
   ]
  },
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    {
     "data": {
      "text/plain": [
       "'SageMath version 8.1.beta8, Release Date: 2017-10-16'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "First we set up the notebook to display mathematical objects using LaTeX rendering:"
   ]
  },
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   "source": [
    "%display latex\n",
    "from sympy import init_printing\n",
    "init_printing()"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Defining a manifold\n",
    "\n",
    "As an example let us define a differentiable manifold of dimension 3 over $\\mathbb{R}$:"
   ]
  },
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   "source": [
    "M = Manifold(3, 'M', latex_name=r'\\mathcal{M}', start_index=1)"
   ]
  },
  {
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   "source": [
    "- The first argument, `3`, is the manifold dimension. In SageManifolds, it can be any\n",
    "  positive integer.\n",
    "- The second argument, `'M'`, is a string defining the manifold's name; it may be \n",
    "  different from the symbol set on the left-hand side of the = sign (here `M`): the latter\n",
    "  stands for a mere Python variable, which refers to the manifold object in the computer \n",
    "  memory, while the string `'M'` is the mathematical symbol chosen for the manifold.\n",
    "- The optional argument `latex_name=r'\\mathcal{M}'` sets the LaTeX\n",
    "  symbol to display the manifold. Note the letter 'r' in front on the first quote: \n",
    "  it indicates that the  string is a *raw* one, so that the backslash character \n",
    "  in `\\mathcal` is considered as an ordinary character (otherwise, the backslash is \n",
    "  used to escape some special characters). If the argument `latex_name` is not \n",
    "  provided by the user, it is set to the string used as the second argument (here `'M'`)\n",
    "- The optional argument `start_index=1` defines the range of indices to be used for \n",
    "  tensor components on the manifold: setting it to 1 means that indices will range \n",
    "  in $\\{1,2,3\\}$. The default value is `start_index=0`.\n",
    "\n",
    "Note that the default base field is $\\mathbb{R}$. If we would have used the optional\n",
    "argument `field='complex'`, we would have defined a manifold over $\\mathbb{C}$. See\n",
    "[list of all options](http://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/manifold.html#sage.manifolds.manifold.Manifold) more details. \n",
    "\n",
    "If we ask for M, it is displayed via its LaTeX symbol:"
   ]
  },
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      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "M"
   ]
  },
  {
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    "deletable": true,
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   "source": [
    "We ask for all symbolic computations to be performed with [SymPy](http://www.sympy.org), instead of SageMath's default symbolic engine (Pynac+Maxima, implemented via SageMath's Symbolic Ring):"
   ]
  },
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   "cell_type": "code",
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   "source": [
    "M.set_calculus_method('sympy')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "If we use the `print` function instead, we get a short description of the object:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Via the command <span style=\"font-family: courier new,courier;\">type</span>, we get the type of the Python object corresponding to M (here the Python class <span style=\"font-family: courier new,courier;\">DifferentiableManifold_with_category</span>):</p>"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
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     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|<class|\\phantom{\\verb!x!}\\verb|'sage.manifolds.differentiable.manifold.DifferentiableManifold_with_category'>|</script></html>"
      ],
      "text/plain": [
       "<class 'sage.manifolds.differentiable.manifold.DifferentiableManifold_with_category'>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(M)"
   ]
  },
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   },
   "source": [
    "We can also ask for the category of M and see that it is the category of smooth manifolds over $\\mathbb{R}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{Smooth}_{\\Bold{R}}</script></html>"
      ],
      "text/plain": [
       "Category of smooth manifolds over Real Field with 53 bits of precision"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "category(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The indices on the manifold are generated by the method <span style=\"font-family: courier new,courier;\">irange()</span>, to be used in loops:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "2\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "for i in M.irange():\n",
    "    print(i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>If the parameter<span style=\"font-family: courier new,courier;\"> start_index</span> had not been specified, the default range of the indices would have been $\\{0,1,2\\}$ instead:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
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    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "1\n",
      "2\n"
     ]
    }
   ],
   "source": [
    "M0 = Manifold(3, 'M', r'\\mathcal{M}')\n",
    "for i in M0.irange():\n",
    "    print(i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Defining a chart on the manifold</h2>\n",
    "<p>Let us assume that the manifold $\\mathcal{M}$ can be covered by a single chart (other cases are discussed below); the chart is declared as follows:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
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   "outputs": [],
   "source": [
    "X.<x,y,z> = M.chart()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The writing `.<x,y,z>` in the left-hand side means that the Python variables `x`, `y` and `z` are set to the three coordinates of the chart. This allows one to refer subsequently to the coordinates by their names.\n",
    "\n",
    "In this example, the function `chart()` has no arguments, which implies that the coordinate symbols will be `x`, `y` and `z` (i.e. exactly the characters set in the `<...>` operator) and that each coordinate range is $(-\\infty,+\\infty)$. For other cases, an argument must be passed to `chart()`  to specify the coordinate symbols and range, as well as the LaTeX symbol of a coordinate if the latter is different from the coordinate name (an example will be provided below)."
   ]
  },
  {
   "cell_type": "markdown",
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   },
   "source": [
    "<p>The chart is displayed as a pair formed by the open set covered by it (here the whole manifold) and the coordinates:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Chart (M, (x, y, z))\n"
     ]
    }
   ],
   "source": [
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(x, y, z)\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (M, (x, y, z))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The coordinates can be accessed individually, by means of their indices, following the convention defined by <span style=\"font-family: courier new,courier;\">start_index=1</span> in the manifold's definition:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x</script></html>"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}y</script></html>"
      ],
      "text/plain": [
       "y"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[2]"
   ]
  },
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   "cell_type": "code",
   "execution_count": 16,
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    "deletable": true,
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}z</script></html>"
      ],
      "text/plain": [
       "z"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[3]"
   ]
  },
  {
   "cell_type": "markdown",
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    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The full set of coordinates is obtained by means of the operator [:]:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(x, y, z\\right)</script></html>"
      ],
      "text/plain": [
       "(x, y, z)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Thanks to the operator `<x,y,z>` used in the chart declaration, each coordinate can be accessed directly via its name:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z is X[3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Coordinates are SageMath symbolic expressions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|<type|\\phantom{\\verb!x!}\\verb|'sage.symbolic.expression.Expression'>|</script></html>"
      ],
      "text/plain": [
       "<type 'sage.symbolic.expression.Expression'>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h3>Functions of the chart coordinates</h3>\n",
    "<p>Real-valued functions of the chart coordinates (mathematically speaking, <em>functions defined on the chart codomain</em>) are generated via the method <span style=\"font-family: courier new,courier;\">function()</span> acting on the chart:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x + y^{2} + z^{3}</script></html>"
      ],
      "text/plain": [
       "x + y**2 + z**3"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = X.function(x+y^2+z^3) ; f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left ( x, \\quad y, \\quad z\\right ) \\mapsto x + y^{2} + z^{3}</script></html>"
      ],
      "text/plain": [
       "(x, y, z) |--> x + y**2 + z**3"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAj0lEQVQIHWNgEFIyYWAQDbZlYGAMYPBP4H7A\noD+Bgf0LA38D3wYGrgUMnCsZ9h/gamDg+8sABP4JPL8hTO4mEJ/jN1CfRjSIOV8BRGpNABJ9IBYD\n12oGBtYAoLwAA/MXBoZSoBD/bxCTJYBBkIHLgIHjI8NVBoZCBrYEBv8CxialuAYGhqmhlgw8////\nbwAA/SMgBrNAMgkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$32$$"
      ],
      "text/plain": [
       "32"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(1,2,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "They belong to SageMath class `ChartFunction`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|<class|\\phantom{\\verb!x!}\\verb|'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>|</script></html>"
      ],
      "text/plain": [
       "<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>and differ from SageMath standard symbolic functions by automatic simplifications in all operations. For instance, adding the two symbolic functions</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "f0(x,y,z) = cos(x)^2 ; g0(x,y,z) = sin(x)^2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>results in</p>"
   ]
  },
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   "cell_type": "code",
   "execution_count": 25,
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y, z \\right) \\ {\\mapsto} \\ \\cos\\left(x\\right)^{2} + \\sin\\left(x\\right)^{2}</script></html>"
      ],
      "text/plain": [
       "(x, y, z) |--> cos(x)^2 + sin(x)^2"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f0 + g0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "while the sum of the corresponding functions in the class `ChartFunction` is automatically simplified:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}1</script></html>"
      ],
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       "1"
      ]
     },
     "execution_count": 26,
     "metadata": {},
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   ],
   "source": [
    "f1 = X.function(cos(x)^2) ; g1 = X.function(sin(x)^2)\n",
    "f1 + g1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>To get the same output with symbolic functions, one has to invoke the method <span style=\"font-family: courier new,courier;\">simplify_trig()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y, z \\right) \\ {\\mapsto} \\ 1</script></html>"
      ],
      "text/plain": [
       "(x, y, z) |--> 1"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(f0 + g0).simplify_trig()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Another difference regards the display; if we ask for the symbolic function f0, we get:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y, z \\right) \\ {\\mapsto} \\ \\cos\\left(x\\right)^{2}</script></html>"
      ],
      "text/plain": [
       "(x, y, z) |--> cos(x)^2"
      ]
     },
     "execution_count": 28,
     "metadata": {},
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    }
   ],
   "source": [
    "f0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>while if we ask for the chart function f1, we get only the coordinate expression:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\cos^{2}{\\left (x \\right )}</script></html>"
      ],
      "text/plain": [
       "cos(x)**2"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>To get an output similar to that of f0, one should call the method <span style=\"font-family: courier new,courier;\">display()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left ( x, \\quad y, \\quad z\\right ) \\mapsto \\cos^{2}{\\left (x \\right )}</script></html>"
      ],
      "text/plain": [
       "(x, y, z) |--> cos(x)**2"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Note that the method `expr()` returns the underlying symbolic expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAZBAMAAACY8Y7RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nY8SXEixf3J17PuYv69hQl+GMuDIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos^{2}{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "cos(x)**2"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1.expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "In the present case, it is a SymPy object:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
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    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|<class|\\phantom{\\verb!x!}\\verb|'sympy.core.power.Pow'>|</script></html>"
      ],
      "text/plain": [
       "<class 'sympy.core.power.Pow'>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(f1.expr())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Introducing a second chart on the manifold\n",
    "\n",
    "Let us first consider an open subset of $\\mathcal{M}$, for instance the complement $U$ of the region defined by $\\{y=0, x\\geq 0\\}$ (note that `(y!=0, x<0)` stands for $y\\not=0$ OR $x<0$; the condition $y\\not=0$ AND $x<0$ would have been written `[y!=0, x<0]` instead):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "U = M.open_subset('U', coord_def={X: (y!=0, x<0)})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Let us call `X_U` the restriction of the chart `X` to the open subset $U$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(x, y, z)\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (U, (x, y, z))"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_U = X.restrict(U) ; X_U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>We introduce another chart on $U$, with spherical-type coordinates $(r,\\theta,\\phi)$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\theta}, {\\phi})\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (U, (r, theta, phi))"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y.<r,th,ph> = U.chart(r'r:(0,+oo) theta:(0,pi):\\theta phi:(0,2*pi):\\phi') ; Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The function <span style=\"font-family: courier new,courier;\">chart()</span> has now some argument; it is a string, which contains specific LaTeX symbols, hence the prefix 'r' to it (for <em>raw</em> string). It also contains the coordinate ranges, since they are different from the default value, which is $(-\\infty, +\\infty)$. For a given coordinate, the various fields are separated by the character ':' and a space character separates the coordinates. Note that for the coordinate $r$, there are only two fields, since the LaTeX symbol has not to be specified. The LaTeX symbols are used for the outputs:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\theta}, {\\phi}\\right)</script></html>"
      ],
      "text/plain": [
       "(theta, phi)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "th, ph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\theta}, {\\phi}\\right)</script></html>"
      ],
      "text/plain": [
       "(theta, phi)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y[2], Y[3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The declared coordinate ranges are now known to Sage, as we may check by means of the command <span style=\"font-family: courier new,courier; font-size: small;\">assumptions()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\verb|x|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|y|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|z|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, \\verb|r|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, r > 0, \\verb|theta|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\theta} > 0, {\\theta} < \\pi, \\verb|phi|\\phantom{\\verb!x!}\\verb|is|\\phantom{\\verb!x!}\\verb|real|, {\\phi} > 0, {\\phi} < 2 \\, \\pi\\right]</script></html>"
      ],
      "text/plain": [
       "[x is real,\n",
       " y is real,\n",
       " z is real,\n",
       " r is real,\n",
       " r > 0,\n",
       " theta is real,\n",
       " theta > 0,\n",
       " theta < pi,\n",
       " phi is real,\n",
       " phi > 0,\n",
       " phi < 2*pi]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
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   "source": [
    "assumptions()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>They are used in simplifications:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r</script></html>"
      ],
      "text/plain": [
       "r"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(abs(r))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left| x \\right|}</script></html>"
      ],
      "text/plain": [
       "abs(x)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(abs(x)) # no simplification occurs since x can take any value in R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>After having been declared, the chart Y can be fully specified by its relation to the chart X_U, via a transition map:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "transit_Y_to_X = Y.transition_map(X_U, [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\theta}, {\\phi})\\right) \\rightarrow \\left(U,(x, y, z)\\right)</script></html>"
      ],
      "text/plain": [
       "Change of coordinates from Chart (U, (r, theta, phi)) to Chart (U, (x, y, z))"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transit_Y_to_X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\\\ y & = & r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\\\ z & = & r \\cos\\left({\\theta}\\right) \\end{array}\\right.</script></html>"
      ],
      "text/plain": [
       "x = r*cos(phi)*sin(theta)\n",
       "y = r*sin(phi)*sin(theta)\n",
       "z = r*cos(theta)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transit_Y_to_X.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The inverse of the transition map can be specified by means of the method <span style=\"font-family: courier new,courier;\">set_inverse()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} r & = & \\sqrt{x^{2} + y^{2} + z^{2}} \\\\ {\\theta} & = & \\arctan\\left(\\sqrt{x^{2} + y^{2}}, z\\right) \\\\ {\\phi} & = & \\arctan\\left(y, x\\right) \\end{array}\\right.</script></html>"
      ],
      "text/plain": [
       "r = sqrt(x^2 + y^2 + z^2)\n",
       "theta = arctan2(sqrt(x^2 + y^2), z)\n",
       "phi = arctan2(y, x)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transit_Y_to_X.set_inverse(sqrt(x^2+y^2+z^2), atan2(sqrt(x^2+y^2),z), atan2(y, x))\n",
    "transit_Y_to_X.inverse().display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>At this stage, the manifold's <strong>atlas</strong> (the \"user atlas\", not the maximal atlas!) contains three charts:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{M},(x, y, z)\\right), \\left(U,(x, y, z)\\right), \\left(U,(r, {\\theta}, {\\phi})\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Chart (M, (x, y, z)), Chart (U, (x, y, z)), Chart (U, (r, theta, phi))]"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.atlas()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The first chart defined on the manifold is considered as the manifold's default chart (it can be changed by the method <span style=\"font-family: courier new,courier;\">set_default_chart()</span>):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M},(x, y, z)\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (M, (x, y, z))"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.default_chart()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Each open subset has its own atlas (since an open subset of a manifold is a manifold by itself):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(U,(x, y, z)\\right), \\left(U,(r, {\\theta}, {\\phi})\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Chart (U, (x, y, z)), Chart (U, (r, theta, phi))]"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.atlas()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(x, y, z)\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (U, (x, y, z))"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.default_chart()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We can draw the chart $Y$ in terms of the chart $X$. \n",
    "Let us first define a viewer for 3D plots (use `'threejs'` or `'jmol'` for interactive 3D graphics):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "viewer3D = 'threejs' # must be 'threejs', 'jmol', 'tachyon' or None (default)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The plot shows lines of constant coordinates from the $Y$ chart in a \"Cartesian frame\" based on the $X$ coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"<!DOCTYPE html>\n",
       "<html>\n",
       "<head>\n",
       "<title></title>\n",
       "<meta charset=&quot;utf-8&quot;>\n",
       "<meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;>\n",
       "<style>\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "  \n",
       "</style>\n",
       "</head>\n",
       "\n",
       "<body>\n",
       "\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/build/three.min.js&quot;></script>\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/examples/js/controls/OrbitControls.js&quot;></script>\n",
       "            \n",
       "<script>\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var options = {&quot;aspect_ratio&quot;: [1.0, 1.0, 1.0], &quot;decimals&quot;: 2, &quot;frame&quot;: true, &quot;axes&quot;: false, &quot;axes_labels&quot;: [&quot;x&quot;, &quot;y&quot;, &quot;z&quot;]};\n",
       "\n",
       "    // When animations are supported by the viewer, the value 'false'\n",
       "    // will be replaced with an option set in Python by the user\n",
       "    var animate = false; // options.animate;\n",
       "\n",
       "    var b = [{&quot;x&quot;:-8.7999998, &quot;y&quot;:-8.7999989, &quot;z&quot;:-8.7999956}, {&quot;x&quot;:8.7999958, &quot;y&quot;:7.999999, &quot;z&quot;:7.999996}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].x - b[0].x, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspect_ratio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "    var autoAspect = 2.5;\n",
       "    if ( zRange > autoAspect * rRange && a[2] === 1 ) a[2] = autoAspect * rRange / zRange;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.LineSegments( box );\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, 'black' ) );\n",
       "\n",
       "    if ( options.axes_labels ) {\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axes_labels;\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + '=' + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + '=' + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + '=' + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z );\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z ) {\n",
       "        var fontsize = 14;\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 32; // powers of two\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.fillStyle = 'black';\n",
       "        context.font = fontsize + 'px monospace';\n",
       "        context.textAlign = 'center';\n",
       "        context.textBaseline = 'middle';\n",
       "        context.fillText( text, .5*canvas.width, .5*canvas.height );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( { map: texture } ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "        sprite.scale.set( 1, .25 ); // ratio of width to height\n",
       "        scene.add( sprite );\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxisHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = new THREE.PerspectiveCamera( 45, window.innerWidth / window.innerHeight, 0.1, 1000 );\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*(xMid+xRange), a[1]*(yMid+yRange), a[2]*(zMid+zRange) );\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "        \n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        camera.aspect = window.innerWidth / window.innerHeight;\n",
       "        camera.updateProjectionMatrix();\n",
       "        if ( !animate ) render();\n",
       "        \n",
       "    } );\n",
       "\n",
       "    var texts = [{&quot;text&quot;:&quot;  x&quot;, &quot;x&quot;:-1.99999983508e-06, &quot;y&quot;:-8.7999989, &quot;z&quot;:-8.7999956},{&quot;text&quot;:&quot;  y&quot;, &quot;x&quot;:8.7999958, &quot;y&quot;:0.0, &quot;z&quot;:-8.7999956},{&quot;text&quot;:&quot;  z&quot;, &quot;x&quot;:-8.7999998, &quot;y&quot;:-8.7999989, &quot;z&quot;:0.0}];\n",
       "    for ( var i=0 ; i < texts.length ; i++ )\n",
       "        addLabel( texts[i].text, a[0]*texts[i].x, a[1]*texts[i].y, a[2]*texts[i].z );\n",
       "\n",
       "    var points = [];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.PointsMaterial( { size: json.size/100, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var lines = [{&quot;points&quot;:[[9.99999333333467e-07, 9.99999666666711e-10, 0.000999999500000042], [0.000109094521864879, 1.09094558229735e-07, 0.109094540047302], [0.000217189044396425, 2.17189116792802e-07, 0.217189080594604], [0.000325283566927971, 3.2528367535587e-07, 0.325283621141905], [0.000433378089459517, 4.33378233918938e-07, 0.433378161689207], [0.000541472611991063, 5.41472792482006e-07, 0.541472702236509], [0.000649567134522609, 6.49567351045074e-07, 0.649567242783811], [0.000757661657054155, 7.57661909608142e-07, 0.757661783331113], [0.000865756179585701, 8.6575646817121e-07, 0.865756323878414], [0.000973850702117247, 9.73851026734278e-07, 0.973850864425716], [0.00108194522464879, 1.08194558529735e-06, 1.08194540497302], [0.00119003974718034, 1.19004014386041e-06, 1.19003994552032], [0.00129813426971188, 1.29813470242348e-06, 1.29813448606762], [0.00140622879224343, 1.40622926098655e-06, 1.40622902661492], [0.00151432331477498, 1.51432381954962e-06, 1.51432356716223], [0.00162241783730652, 1.62241837811268e-06, 1.62241810770953], [0.00173051235983807, 1.73051293667575e-06, 1.73051264825683], [0.00183860688236961, 1.83860749523882e-06, 1.83860718880413], [0.00194670140490116, 1.94670205380189e-06, 1.94670172935143], [0.00205479592743271, 2.05479661236496e-06, 2.05479626989873], [0.00216289044996425, 2.16289117092802e-06, 2.16289081044604], [0.0022709849724958, 2.27098572949109e-06, 2.27098535099334], [0.00237907949502734, 2.37908028805416e-06, 2.37907989154064], [0.00248717401755889, 2.48717484661723e-06, 2.48717443208794], [0.00259526854009043, 2.59526940518029e-06, 2.59526897263524], [0.00270336306262198, 2.70336396374336e-06, 2.70336351318254], [0.00281145758515353, 2.81145852230643e-06, 2.81145805372984], [0.00291955210768507, 2.9195530808695e-06, 2.91955259427715], [0.00302764663021662, 3.02764763943256e-06, 3.02764713482445], [0.00313574115274816, 3.13574219799563e-06, 3.13574167537175], [0.00324383567527971, 3.2438367565587e-06, 3.24383621591905], [0.00335193019781125, 3.35193131512177e-06, 3.35193075646635], [0.0034600247203428, 3.46002587368484e-06, 3.46002529701365], [0.00356811924287435, 3.5681204322479e-06, 3.56811983756096], [0.00367621376540589, 3.67621499081097e-06, 3.67621437810826], [0.00378430828793744, 3.78430954937404e-06, 3.78430891865556], [0.00389240281046898, 3.89240410793711e-06, 3.89240345920286], [0.00400049733300053, 4.00049866650017e-06, 4.00049799975016], [0.00410859185553207, 4.10859322506324e-06, 4.10859254029746], [0.00421668637806362, 4.21668778362631e-06, 4.21668708084476], [0.00432478090059517, 4.32478234218938e-06, 4.32478162139207], [0.00443287542312671, 4.43287690075244e-06, 4.43287616193937], [0.00454096994565826, 4.54097145931551e-06, 4.54097070248667], [0.0046490644681898, 4.64906601787858e-06, 4.64906524303397], [0.00475715899072135, 4.75716057644165e-06, 4.75715978358127], [0.00486525351325289, 4.86525513500471e-06, 4.86525432412857], [0.00497334803578444, 4.97334969356778e-06, 4.97334886467588], [0.00508144255831599, 5.08144425213085e-06, 5.08144340522318], [0.00518953708084753, 5.18953881069392e-06, 5.18953794577048], [0.00529763160337908, 5.29763336925698e-06, 5.29763248631778], [0.00540572612591062, 5.40572792782005e-06, 5.40572702686508], [0.00551382064844217, 5.51382248638312e-06, 5.51382156741238], [0.00562191517097371, 5.62191704494619e-06, 5.62191610795969], [0.00573000969350526, 5.73001160350926e-06, 5.73001064850699], [0.00583810421603681, 5.83810616207232e-06, 5.83810518905429], [0.00594619873856835, 5.94620072063539e-06, 5.94619972960159], [0.0060542932610999, 6.05429527919846e-06, 6.05429427014889], [0.00616238778363144, 6.16238983776153e-06, 6.16238881069619], [0.00627048230616299, 6.27048439632459e-06, 6.2704833512435], [0.00637857682869453, 6.37857895488766e-06, 6.3785778917908], [0.00648667135122608, 6.48667351345073e-06, 6.4866724323381], [0.00659476587375763, 6.5947680720138e-06, 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[0.00475254367656354, -0.00643531718479906, -7.99999600000033], [0.00528101319689937, -0.00600923275308828, -7.99999600000033], [0.00577145710100758, -0.0055398792042646, -7.99999600000033], [0.00622034397720949, -0.00503063609017864, -7.99999600000033], [0.00662444164269328, -0.00448517018508389, -7.99999600000033], [0.00698084041665087, -0.00390740908325043, -7.99999600000033], [0.0072869740712587, -0.00330151291855987, -7.99999600000033], [0.00754063830964598, -0.00267184440971281, -7.99999600000033], [0.00774000663780108, -0.00202293744673666, -7.99999600000033], [0.00788364351613116, -0.00135946444498461, -7.99999600000033], [0.00797051469597828, -0.000686202701691374, -7.99999600000033], [0.00799999466666428, -7.99999733327802e-06, -7.99999600000033]], &quot;color&quot;:&quot;red&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:1}];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length - 1 ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "            var v = json.points[i+1];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.LineBasicMaterial( { color: json.color, linewidth: json.linewidth,\n",
       "                                                      transparent: transparent, opacity: json.opacity } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.LineSegments( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var surfaces = [];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                geometry.faces.push( new THREE.Face3( f[0], f[j+1], f[j+2] ) );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.MeshPhongMaterial( {\n",
       "                                     color: json.color, side: THREE.DoubleSide,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var scratch = new THREE.Vector3();\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "        for ( var i=0 ; i < scene.children.length ; i++ ) {\n",
       "            if ( scene.children[i].type === 'Sprite' ) {\n",
       "                var sprite = scene.children[i];\n",
       "                var adjust = scratch.addVectors( sprite.position, scene.position )\n",
       "                               .sub( camera.position ).length() / 5;\n",
       "                sprite.scale.set( adjust, .25*adjust ); // ratio of canvas width to height\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "</script>\n",
       "\n",
       "</body>\n",
       "</html>\" \n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "graph = Y.plot(X)\n",
    "show(graph, viewer=viewer3D, online=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The command plot() allows for many options, to control the number of coordinate lines to be drawn, their style and color, as well as the coordinate ranges (cf. the [list of all options](http://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/chart.html#sage.manifolds.chart.RealChart.plot)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"<!DOCTYPE html>\n",
       "<html>\n",
       "<head>\n",
       "<title></title>\n",
       "<meta charset=&quot;utf-8&quot;>\n",
       "<meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;>\n",
       "<style>\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "  \n",
       "</style>\n",
       "</head>\n",
       "\n",
       "<body>\n",
       "\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/build/three.min.js&quot;></script>\n",
       "<script src=&quot;https://cdn.rawgit.com/mrdoob/three.js/r80/examples/js/controls/OrbitControls.js&quot;></script>\n",
       "            \n",
       "<script>\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var options = {&quot;aspect_ratio&quot;: [1.0, 1.0, 1.0], &quot;decimals&quot;: 2, &quot;frame&quot;: true, &quot;axes&quot;: false, &quot;axes_labels&quot;: [&quot;x&quot;, &quot;y&quot;, &quot;z&quot;]};\n",
       "\n",
       "    // When animations are supported by the viewer, the value 'false'\n",
       "    // will be replaced with an option set in Python by the user\n",
       "    var animate = false; // options.animate;\n",
       "\n",
       "    var b = [{&quot;x&quot;:-2.19999995, &quot;y&quot;:-2.19806538293, &quot;z&quot;:-0.1}, {&quot;x&quot;:2.19999895, &quot;y&quot;:1.99824125721, &quot;z&quot;:2.0}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].x - b[0].x, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspect_ratio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "    var autoAspect = 2.5;\n",
       "    if ( zRange > autoAspect * rRange && a[2] === 1 ) a[2] = autoAspect * rRange / zRange;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.LineSegments( box );\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, 'black' ) );\n",
       "\n",
       "    if ( options.axes_labels ) {\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axes_labels;\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + '=' + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + '=' + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + '=' + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z );\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z ) {\n",
       "        var fontsize = 14;\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 32; // powers of two\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.fillStyle = 'black';\n",
       "        context.font = fontsize + 'px monospace';\n",
       "        context.textAlign = 'center';\n",
       "        context.textBaseline = 'middle';\n",
       "        context.fillText( text, .5*canvas.width, .5*canvas.height );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( { map: texture } ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "        sprite.scale.set( 1, .25 ); // ratio of width to height\n",
       "        scene.add( sprite );\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxisHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = new THREE.PerspectiveCamera( 45, window.innerWidth / window.innerHeight, 0.1, 1000 );\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*(xMid+xRange), a[1]*(yMid+yRange), a[2]*(zMid+zRange) );\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i < lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === 'camera' ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( 'change', function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( 'resize', function() {\n",
       "        \n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        camera.aspect = window.innerWidth / window.innerHeight;\n",
       "        camera.updateProjectionMatrix();\n",
       "        if ( !animate ) render();\n",
       "        \n",
       "    } );\n",
       "\n",
       "    var texts = [{&quot;text&quot;:&quot;  x&quot;, &quot;x&quot;:-4.99999954995e-07, &quot;y&quot;:-2.19806538293, &quot;z&quot;:-0.1},{&quot;text&quot;:&quot;  y&quot;, &quot;x&quot;:2.19999895, &quot;y&quot;:0.0, &quot;z&quot;:-0.1},{&quot;text&quot;:&quot;  z&quot;, &quot;x&quot;:-2.19999995, &quot;y&quot;:-2.19806538293, &quot;z&quot;:1.0}];\n",
       "    for ( var i=0 ; i < texts.length ; i++ )\n",
       "        addLabel( texts[i].text, a[0]*texts[i].x, a[1]*texts[i].y, a[2]*texts[i].z );\n",
       "\n",
       "    var points = [];\n",
       "    for ( var i=0 ; i < points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( 'canvas' );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( '2d' );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.PointsMaterial( { size: json.size/100, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var lines = [{&quot;points&quot;:[[0.0, 0.0, 1.0], [0.0, 0.0, 1.01351351351351], [0.0, 0.0, 1.02702702702703], [0.0, 0.0, 1.04054054054054], [0.0, 0.0, 1.05405405405405], [0.0, 0.0, 1.06756756756757], [0.0, 0.0, 1.08108108108108], [0.0, 0.0, 1.0945945945946], [0.0, 0.0, 1.10810810810811], [0.0, 0.0, 1.12162162162162], [0.0, 0.0, 1.13513513513514], [0.0, 0.0, 1.14864864864865], [0.0, 0.0, 1.16216216216216], [0.0, 0.0, 1.17567567567568], [0.0, 0.0, 1.18918918918919], [0.0, 0.0, 1.2027027027027], [0.0, 0.0, 1.21621621621622], [0.0, 0.0, 1.22972972972973], [0.0, 0.0, 1.24324324324324], [0.0, 0.0, 1.25675675675676], [0.0, 0.0, 1.27027027027027], [0.0, 0.0, 1.28378378378379], [0.0, 0.0, 1.2972972972973], [0.0, 0.0, 1.31081081081081], [0.0, 0.0, 1.32432432432433], [0.0, 0.0, 1.33783783783784], [0.0, 0.0, 1.35135135135135], [0.0, 0.0, 1.36486486486487], [0.0, 0.0, 1.37837837837838], [0.0, 0.0, 1.39189189189189], [0.0, 0.0, 1.40540540540541], [0.0, 0.0, 1.41891891891892], [0.0, 0.0, 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[1.82174382143942, -0.825378367203377, 1.22464679914735e-16], [1.88515989160561, -0.667961213755353, 1.22464679914735e-16], [1.93500198195142, -0.505734445973454, 1.22464679914735e-16], [1.97091120751883, -0.339866167890659, 1.22464679914735e-16], [1.99262900610025, -0.1715507040147, 1.22464679914735e-16], [1.99999900000008, -0.00199999966665363, 1.22464679914735e-16]], &quot;color&quot;:&quot;red&quot;, &quot;opacity&quot;:1, &quot;linewidth&quot;:1}];\n",
       "    for ( var i=0 ; i < lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.points.length - 1 ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "            var v = json.points[i+1];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.LineBasicMaterial( { color: json.color, linewidth: json.linewidth,\n",
       "                                                      transparent: transparent, opacity: json.opacity } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.LineSegments( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var surfaces = [];\n",
       "    for ( var i=0 ; i < surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i < json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i < json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j < f.length - 2 ; j++ ) {\n",
       "                geometry.faces.push( new THREE.Face3( f[0], f[j+1], f[j+2] ) );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var transparent = json.opacity < 1 ? true : false;\n",
       "        var material = new THREE.MeshPhongMaterial( {\n",
       "                                     color: json.color, side: THREE.DoubleSide,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20 } );\n",
       "\n",
       "        var c = geometry.center().multiplyScalar( -1 );\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        scene.add( mesh );\n",
       "    }\n",
       "\n",
       "    var scratch = new THREE.Vector3();\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "        for ( var i=0 ; i < scene.children.length ; i++ ) {\n",
       "            if ( scene.children[i].type === 'Sprite' ) {\n",
       "                var sprite = scene.children[i];\n",
       "                var adjust = scratch.addVectors( sprite.position, scene.position )\n",
       "                               .sub( camera.position ).length() / 5;\n",
       "                sprite.scale.set( adjust, .25*adjust ); // ratio of canvas width to height\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "</script>\n",
       "\n",
       "</body>\n",
       "</html>\" \n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "graph = Y.plot(X, ranges={r:(1,2), th:(0,pi/2)}, number_values=4, \n",
    "               color={r:'blue', th:'green', ph:'red'})\n",
    "show(graph, aspect_ratio=1, viewer=viewer3D, online=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Conversly, the chart $X|_{U}$ can be plotted in terms of the chart $Y$ (this is not possible for the whole chart $X$ since its domain is larger than that of chart $Y$):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "#graph = X_U.plot(Y)\n",
    "#show(graph, viewer=viewer3D, online=True, axes_labels=['r','theta','phi'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Points on the manifold</h2>\n",
    "<p>A point on $\\mathcal{M}$ is defined by its coordinates in a given chart:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Point p on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}p</script></html>"
      ],
      "text/plain": [
       "Point p on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = M.point((1,2,-1), chart=X, name='p') ; print(p) ; p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Since $X=(\\mathcal{M}, (x,y,z))$ is the manifold's default chart, its name can be omitted:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Point p on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}p</script></html>"
      ],
      "text/plain": [
       "Point p on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = M.point((1,2,-1), name='p') ; print(p) ; p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Of course, $p$ belongs to $\\mathcal{M}$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p in M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>It is also in $U$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p in U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Indeed the coordinates of $p$ have $y\\not=0$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1, 2, -1\\right)</script></html>"
      ],
      "text/plain": [
       "(1, 2, -1)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.coord(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Note in passing that since $X$ is the default chart on $\\mathcal{M}$, its name can be omitted in the arguments of coord():</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1, 2, -1\\right)</script></html>"
      ],
      "text/plain": [
       "(1, 2, -1)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.coord()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The coordinates of $p$ can also be obtained by letting the chart acting of the point (from the very definition of a chart!):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1, 2, -1\\right)</script></html>"
      ],
      "text/plain": [
       "(1, 2, -1)"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let $q$ be a point with $y = 0$ and $x \\geq 0$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "q = M.point((1,0,2), name='q')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>This time, the point does not belong to $U$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{False}</script></html>"
      ],
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q in U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Accordingly, we cannot ask for the coordinates of $q$ in the chart $Y=(U, (r,\\theta,\\phi))$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error: the point does not belong to the domain of Chart (U, (r, theta, phi))\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    q.coord(Y)\n",
    "except ValueError as exc:\n",
    "    print(\"Error: \" + str(exc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>but we can for point $p$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUsAAAAmBAMAAAC/hKSdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left ( \\sqrt{6}, \\quad - \\operatorname{atan}{\\left (\\sqrt{5} \\right )} + \\pi, \\quad \\operatorname{atan}{\\left (2 \\right )}\\right )$$"
      ],
      "text/plain": [
       "(sqrt(6), -atan(sqrt(5)) + pi, atan(2))"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.coord(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUsAAAAmBAMAAAC/hKSdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left ( \\sqrt{6}, \\quad - \\operatorname{atan}{\\left (\\sqrt{5} \\right )} + \\pi, \\quad \\operatorname{atan}{\\left (2 \\right )}\\right )$$"
      ],
      "text/plain": [
       "(sqrt(6), -atan(sqrt(5)) + pi, atan(2))"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Points can be compared:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{False}</script></html>"
      ],
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q == p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = U.point((sqrt(6), pi-atan(sqrt(5)), atan(2)), Y)\n",
    "p1 == p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In SageMath's terminology, points are <strong>elements</strong>, whose <strong>parents</strong> are the manifold on which they have been defined:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}U</script></html>"
      ],
      "text/plain": [
       "Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1.parent()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Scalar fields</h2>\n",
    "<p>A scalar field is a differentiable mapping $U \\longrightarrow \\mathbb{R}$, where $U$ is an open subset of $\\mathcal{M}$.</p>\n",
    "<p>The scalar field is defined by its expressions in terms of charts covering its domain (in general more than one chart is necessary to cover all the domain):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field f on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "f = U.scalar_field({X_U: x+y^2+z^3}, name='f') ; print(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The coordinate expressions of the scalar field are passed as a Python dictionary, with the charts as keys, hence the writing <span style=\"font-family: courier new,courier;\">{X_U: x+y^2+z^3}</span>.</p>\n",
    "<p>Since in the present case, there is only one chart in the dictionary, an alternative writing is</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field f on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "f = U.scalar_field(x+y^2+z^3, chart=X_U, name='f') ; print(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Since X_U is the domain's default chart, it can be omitted in the above declaration:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field f on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "f = U.scalar_field(x+y^2+z^3, name='f') ; print(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "As a mapping $U\\subset\\mathcal{M}\\longrightarrow\\mathbb{R}$, a scalar field acts on points, not on coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPBAMAAAAv0UM9AAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAMpndu3bvImbNiRBUq9OB\nhjcAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABESURBVAgdY2BgYBACYgYGExDBmgIiK6aAyAUgkqMA\nRG5lAJELQCSPAIjcxQAiz969++wqUIIBrIvhCYi55N0NEMXAAABbkhBrtxdTYQAAAABJRU5ErkJg\ngg==\n",
      "text/latex": [
       "$$4$$"
      ],
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The method `display()` provides the expression of the scalar field in terms of a given chart:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x + y^{2} + z^{3} \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: U --> R\n",
       "   (x, y, z) |--> x + y**2 + z**3"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display(X_U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "If no argument is provided, the method `display()` shows the coordinate expression of the scalar field in all the charts defined on the domain (except for *subcharts*, i.e. the restrictions of some chart to a subdomain):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x + y^{2} + z^{3} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: U --> R\n",
       "   (x, y, z) |--> x + y**2 + z**3\n",
       "   (r, theta, phi) |--> r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Note that the expression of $f$ in terms of the coordinates $(r,\\theta,\\phi)$ has not been provided by the user but has been automatically computed by means of the change-of-coordinate formula declared above in the transition map. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: U --> R\n",
       "   (r, theta, phi) |--> r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display(Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "In each chart, the scalar field is represented by a function of the chart coordinates (an object of the type `ChartFunction` described above), which is accessible via the method `coord_function()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x + y^{2} + z^{3}</script></html>"
      ],
      "text/plain": [
       "x + y**2 + z**3"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.coord_function(X_U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left ( x, \\quad y, \\quad z\\right ) \\mapsto x + y^{2} + z^{3}</script></html>"
      ],
      "text/plain": [
       "(x, y, z) |--> x + y**2 + z**3"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.coord_function(X_U).display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right)</script></html>"
      ],
      "text/plain": [
       "r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.coord_function(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left ( r, \\quad theta, \\quad phi\\right ) \\mapsto r \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right)</script></html>"
      ],
      "text/plain": [
       "(r, theta, phi) |--> r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.coord_function(Y).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The \"raw\" symbolic expression (here a SymPy object) is returned by the method `expr()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGYAAAAYBAMAAAD62ZXgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$x + y^{2} + z^{3}$$"
      ],
      "text/plain": [
       "x + y**2 + z**3"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expr(X_U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAbUAAAAcBAMAAAAU4Dx8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARImrEHa7zVTvMt2Z\nImbh7FZmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGLUlEQVRYCdVYXYhUZRh+Zjw7szuzMzt4Ef1c\nOCriRSob2h9UTgZZYTkIYsJWe5F1kzoVsRlEp4uiKNq1vAjJHIPIpWC2IjJobVhc7YdiLiKyxB0E\ni0DzNxOR7Hm/7zvnfHP2zOyxrnxhz/f+PO/zfu/5fmZmgbAk9+8Lu5AcWDbNN5MjiieUEwMSyvif\n5mLcPI3hB+yf5pvJEcUTyokBCWX8V3OXThxHrRGmmMCJsGtGO4onlBQDEsqIbXa1IDOD2vwYU4WW\ngDIuf92ieVqYY0Ba8JdhJPptcLrkWcO+5nngbPfV+EoETzg5BiScEtNeYuMmfeMzX/OVdWWl5pqA\n8yfy0zatD7QVzXNYuRrRSRoSl1ZweKCJ43aVQFdhmErZkhU46+nJuqdZo96TP9Fz1T113G5FQuqo\nfgv0ap7coAIUESSFIehEG4ChcIlHX8es6HcrNICp1FUXQ0vQ0THPZY9TrljPA5ltOIuVdqhVrxU9\nW/Ok9Tzo9ZPCkI60AViVx2ipr+RUvBotI2dHKepKuZPKUo8e1+iJYqJqVG/gaZtq0sjXgWwT55Es\nm9BLZowYDI/u0GGCn+SDDSQureCwF30uFvgUlqLC8CvtDUK1ktE/eOVlT/Wj41gvvmwD+AP4B86I\nCXXozfCMK6S8Lj/J5AIGEpdWcPkR9DVxr09hKRIG/ErWzc45a9l56ZKn+uN7a9QJ+5WOL5E5Dzxp\nQtN7+6VqQponN6bMpDy9pBAEHWl9MBQuW0fNRbpgitiD0HB3yEMqrWef8++cV6Z6gH+zl832VtuZ\nWORixdq5yDx4tMKQkgHCT6HrFLDIeExvhuT9N9blNvdntny79kYThz7Suw793NBJERC0pQ2BicOc\nAh4p6Gtq1byv4SxYUwBhUk7CCCrNKSGZejY9Qudj/Cs+7m4RAOWgm9mRGUeqfDcwpl3AJmDWmaW3\nngR+Ny7TmyZxBpFFuh/ZDZhiK0pSTQ697mG5L5gUBWlLGwazPJYvXbqTO7MC5N7GaOPDBjYoGEMS\nDiqhVsXubL+6O57mFeh+imsJEBlC94VsEZkL15RQ1i5gD49cBX0kfs24TG+aJLENXUj1I6XOhEZk\nXY6b+Ik0XyVFQdrShsEsj8+Bv4HuMSBdwXD1E6C2S8pSJBxUkm9XpdUFCfAcwXG427Q4FzlyY+N0\n9txDxgdsBfqKyl0T38KhoS+GhvhSPJKvNjRUb+zfex89ZM+P8WL+jnxERkDa04bALI/T6rhnTgKr\n61wNHv2+osAoDNuVplxgWPyATLGbKVoSf3Ec5rzOOQsv1Y1Tsle7+Iam6o2jWTdDsuKFi6q3wdbe\nUnX2Rn5JioC0pw2BWd45j94R9sSJzilzwqfY26DAKAynrEryrfiA+HVvvRWtk8Nbt4s/Ik8CLVx1\nrrK0fZ3xeL0pklwJJwqyJ63esi7Q08w1c3zDTIqCyGaKpA2DiePOki2RGLPW7Xspy/kwbFVSv2be\n1NO8iUNPUet8DvGl9NR5LXL2d3hentaeQpK84bsEiqS7jFnNUG9yl6TL+UaqrpKiIHIJRNKGwXJZ\nbMX1fOYr6rzBfYfrd7WUpZNhqxJ4T+bMqvDjwP5xc7CJxZntSBdfLWEjQ0p48/eWDzaoTxiPWTdN\n0r0NqQYXzV63ZF3echJP6KQoiHygRNKGwfLBM5nbwWdvkVvgXSQbo1U8pWB0MmxVki8ZsrwibNPf\naTSd29Y0cGjeDZh99EhB4iLcVM7kPtGekwfF9KZJEksenptZfua+5Wd2b35LxzX9R78NVGkzKQrS\nljYMloOwaqDEZ7rKx12Ti5GbmF8QGE01+6CS/Z8CudBmkqwsmYj5tuH3pr2Rz3HxHpaHnySGLXFp\nfZz6hmIzKF2Fg0rPBABZ5plEfxslyv/ae/9MKTgmiOPy8JPEsCUurY/r8F3Zr+T4v9l4rY7Y5dro\nG41/ZZt4hDsla12UQPukuLQezqkI3zSRcFG8UinfL5qRWzylw6h//SG4XTpgTSg3SKUphjcz0Vsl\nLq2HU++rlUIsCTdFkUrZqmhG2r9WD8Ej01R6XtYirqgTQHCHpLi0Btf5fwqm0hF7ft1127rSdbVd\ngiZeDNQrX+stXfk9dOrgXy6YB2cihDC2AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$r \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right)$$"
      ],
      "text/plain": [
       "r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expr(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expr(Y) is f.coord_function(Y).expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A scalar field can also be defined by some unspecified function of the coordinates:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field h on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "h = U.scalar_field(function('H')(x, y, z), name='h') ; print(h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} h:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & H{\\left (x,y,z \\right )} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & H{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "h: U --> R\n",
       "   (x, y, z) |--> H(x, y, z)\n",
       "   (r, theta, phi) |--> H(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} h:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & H{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "h: U --> R\n",
       "   (r, theta, phi) |--> H(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h.display(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAUBAMAAABrMp7fAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$H{\\left (1,2,-1 \\right )}$$"
      ],
      "text/plain": [
       "H(1, 2, -1)"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(p) # remember that p is the point of coordinates (1,2,-1) in the chart X_U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The parent of $f$ is the set $C^\\infty(U)$ of all smooth scalar fields on $U$, which is a commutative algebra over $\\mathbb{R}$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}C^{\\infty}\\left(U\\right)</script></html>"
      ],
      "text/plain": [
       "Algebra of differentiable scalar fields on the Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CU = f.parent() ; CU"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Algebra of differentiable scalar fields on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(CU)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbf{CommutativeAlgebras}_{\\text{SR}}</script></html>"
      ],
      "text/plain": [
       "Category of commutative algebras over Symbolic Ring"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CU.category()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<div id=\"insert_new_cell_before91\" class=\"insert_new_cell\"> </div>\n",
    "<div id=\"cell_text_91\" class=\"text_cell\">\n",
    "<p>The base ring of the algebra is the field $\\mathbb{R}$, which is represented here by SageMath's Symbolic Ring (SR):</p>\n",
    "</div>\n",
    "<div id=\"insert_new_cell_before86\" class=\"insert_new_cell\"> </div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\text{SR}</script></html>"
      ],
      "text/plain": [
       "Symbolic Ring"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CU.base_ring()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Arithmetic operations on scalar fields are defined <span id=\"cell_outer_92\"><span id=\"cell_outer_91\"><span id=\"cell_outer_92\">through the algebra structure:</span></span></span></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "s = f + 2*h ; print(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x + y^{2} + z^{3} + 2 H{\\left (x,y,z \\right )} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{3} \\cos^{3}{\\left (\\theta \\right )} + r^{2} \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 2 H{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "U --> R\n",
       "(x, y, z) |--> x + y**2 + z**3 + 2*H(x, y, z)\n",
       "(r, theta, phi) |--> r**3*cos(theta)**3 + r**2*sin(phi)**2*sin(theta)**2 + r*sin(theta)*cos(phi) + 2*H(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Tangent spaces</h2>\n",
    "<p>The tangent vector space to the manifold at point $p$ is obtained as follows:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T_{p}\\,\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "Tangent space at Point p on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tp = M.tangent_space(p) ; Tp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangent space at Point p on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(Tp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>$T_p\\, \\mathcal{M}$ is a 2-dimensional vector space over $\\mathbb{R}$ (represented here by SageMath's Symbolic Ring (SR)) :</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Category of finite dimensional vector spaces over Symbolic Ring\n"
     ]
    }
   ],
   "source": [
    "print(Tp.category())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}3</script></html>"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tp.dim()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>$T_p\\, \\mathcal{M}$ is automatically endowed with vector bases deduced from the vector frames defined around the point:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y },\\frac{\\partial}{\\partial z }\\right), \\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Basis (d/dx,d/dy,d/dz) on the Tangent space at Point p on the 3-dimensional differentiable manifold M,\n",
       " Basis (d/dr,d/dtheta,d/dphi) on the Tangent space at Point p on the 3-dimensional differentiable manifold M]"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tp.bases()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>For the tangent space at the point $q$, on the contrary, there is only one pre-defined basis, since $q$ is not in the domain $U$ of the frame associated with coordinates $(r,\\theta,\\phi)$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y },\\frac{\\partial}{\\partial z }\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Basis (d/dx,d/dy,d/dz) on the Tangent space at Point q on the 3-dimensional differentiable manifold M]"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tq = M.tangent_space(q)\n",
    "Tq.bases()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A random element:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangent vector at Point p on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "v = Tp.an_element() ; print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial x } + 2 \\frac{\\partial}{\\partial y } + 3 \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "d/dx + 2 d/dy + 3 d/dz"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangent vector at Point q on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "u = Tq.an_element() ; print(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial x } + 2 \\frac{\\partial}{\\partial y } + 3 \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "d/dx + 2 d/dy + 3 d/dz"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Note that, despite what the above simplified writing may suggest (the mention of the point $p$ or $q$ is omitted in the basis vectors), $u$ and $v$ are different vectors, for they belong to different vector spaces:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T_{p}\\,\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "Tangent space at Point p on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T_{q}\\,\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "Tangent space at Point q on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.parent()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In particular, it is not possible to add $u$ and $v$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Error: unsupported operand parent(s) for +: 'Tangent space at Point q on the 3-dimensional differentiable manifold M' and 'Tangent space at Point p on the 3-dimensional differentiable manifold M'\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    s = u + v\n",
    "except TypeError as exc:\n",
    "    print(\"Error: \" + str(exc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Vector Fields</h2>\n",
    "<p>Each chart defines a vector frame on the chart domain: the so-called <strong>coordinate basis</strong>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y },\\frac{\\partial}{\\partial z }\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "Coordinate frame (M, (d/dx,d/dy,d/dz))"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.frame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.frame().domain()  # this frame is defined on the whole manifold"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U, \\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "Coordinate frame (U, (d/dr,d/dtheta,d/dphi))"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y.frame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}U</script></html>"
      ],
      "text/plain": [
       "Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y.frame().domain() # this frame is defined only on U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The list of frames defined on a given open subset is <span>returned by the method <span style=\"font-family: courier new,courier;\">frames()</span></span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y },\\frac{\\partial}{\\partial z }\\right)\\right), \\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y },\\frac{\\partial}{\\partial z }\\right)\\right), \\left(U, \\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Coordinate frame (M, (d/dx,d/dy,d/dz)),\n",
       " Coordinate frame (U, (d/dx,d/dy,d/dz)),\n",
       " Coordinate frame (U, (d/dr,d/dtheta,d/dphi))]"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.frames()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(U, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y },\\frac{\\partial}{\\partial z }\\right)\\right), \\left(U, \\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Coordinate frame (U, (d/dx,d/dy,d/dz)),\n",
       " Coordinate frame (U, (d/dr,d/dtheta,d/dphi))]"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.frames()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}, \\left(\\frac{\\partial}{\\partial x },\\frac{\\partial}{\\partial y },\\frac{\\partial}{\\partial z }\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "Coordinate frame (M, (d/dx,d/dy,d/dz))"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.default_frame()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Unless otherwise specified (via the command <span style=\"font-family: courier new,courier;\">set_default_frame()</span>), the default frame is that associated with the default chart:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.default_frame() is M.default_chart().frame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.default_frame() is U.default_chart().frame()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Individual elements of a frame can be accessed by means of their indices:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial y }</script></html>"
      ],
      "text/plain": [
       "Vector field d/dy on the Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = U.default_frame() ; e2 = e[2] ; e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vector field d/dy on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(e2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p><span id=\"cell_outer_109\">We may define a new vector field as follows:</span></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vector field on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "v = e[2] + 2*x*e[3] ; print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial y } + 2 x \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "d/dy + 2*x d/dz"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A vector field can be defined by its components with respect to a given vector frame. When the latter is not specified, the open set's default frame is of course assumed:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\left( y + 1 \\right) \\frac{\\partial}{\\partial x } - x \\frac{\\partial}{\\partial y } + x y z \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "v = (y + 1) d/dx - x d/dy + x*y*z d/dz"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = U.vector_field(name='v') # vector field defined on the open set U\n",
    "v[1] = 1+y\n",
    "v[2] = -x\n",
    "v[3] = x*y*z\n",
    "v.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Vector fields on $U$ are Sage <em>element</em> objects, whose <em>parent</em> is the set $\\mathfrak{X}(U)$ of vector fields defined on $U$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathfrak{X}\\left(U\\right)</script></html>"
      ],
      "text/plain": [
       "Free module X(U) of vector fields on the Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.parent()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The set $\\mathfrak{X}(U)$ is a module over the commutative algebra $C^\\infty(U)$ of scalar fields on $U$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Free module X(U) of vector fields on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(v.parent())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Category of finite dimensional modules over Algebra of differentiable scalar fields on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(v.parent().category())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}C^{\\infty}\\left(U\\right)</script></html>"
      ],
      "text/plain": [
       "Algebra of differentiable scalar fields on the Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.parent().base_ring()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A vector field acts on scalar fields:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x + y^{2} + z^{3} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: U --> R\n",
       "   (x, y, z) |--> x + y**2 + z**3\n",
       "   (r, theta, phi) |--> r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field v(f) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "s = v(f) ; print(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} v\\left(f\\right):& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & 3 x y z^{3} - 2 x y + y + 1 \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & 3 r^{5} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} + 3 r^{5} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{5}{\\left (\\theta \\right )} - 2 r^{2} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + 1 \\end{array}</script></html>"
      ],
      "text/plain": [
       "v(f): U --> R\n",
       "   (x, y, z) |--> 3*x*y*z**3 - 2*x*y + y + 1\n",
       "   (r, theta, phi) |--> 3*r**5*sin(phi)*sin(theta)**4*cos(phi)*cos(theta)**3 + 3*r**5*sin(phi)*sin(theta)**2*cos(phi)*cos(theta)**5 - 2*r**2*sin(phi)*sin(theta)**2*cos(phi) + r*sin(phi)*sin(theta) + 1"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial z } = \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "d/dz = d/dz"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e[3].display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\frac{\\partial}{\\partial z } \\left( f \\right) : & U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & 3 z^{2} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & 3 r^{2} \\cos^{2}{\\left (\\theta \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "d/dz(f): U --> R\n",
       "   (x, y, z) |--> 3*z**2\n",
       "   (r, theta, phi) |--> 3*r**2*cos(theta)**2"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e[3](f).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Unset components are assumed to be zero:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}w = 3 \\frac{\\partial}{\\partial y }</script></html>"
      ],
      "text/plain": [
       "w = 3 d/dy"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = U.vector_field(name='w')\n",
    "w[2] = 3\n",
    "w.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A vector field on $U$ can be expanded in the vector frame associated with the chart $(r,\\theta,\\phi)$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\left( \\frac{x \\left(y z^{2} + 1\\right)}{\\sqrt{x^{2} + y^{2} + z^{2}}} \\right) \\frac{\\partial}{\\partial r } + \\left( \\frac{x z \\left(- x^{2} y - y^{3} + 1\\right)}{\\sqrt{x^{2} + y^{2}} \\left(x^{2} + y^{2} + z^{2}\\right)} \\right) \\frac{\\partial}{\\partial {\\theta} } + \\left( - \\frac{x^{2} + y^{2} + y}{x^{2} + y^{2}} \\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = x*(y*z**2 + 1)/sqrt(x**2 + y**2 + z**2) d/dr + x*z*(-x**2*y - y**3 + 1)/(sqrt(x**2 + y**2)*(x**2 + y**2 + z**2)) d/dtheta - (x**2 + y**2 + y)/(x**2 + y**2) d/dphi"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display(Y.frame())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>By default, the components are expressed in terms of the default coordinates $(x,y,z)$. To express them in terms of the coordinates $(r,\\theta,\\phi)$, one should add the corresponding chart as the second argument of the method <span style=\"font-family: courier new,courier;\">display()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\left(r^{3} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos^{2}{\\left (\\theta \\right )} + 1\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\frac{\\partial}{\\partial r } + \\frac{1}{r} \\left(- r^{3} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + 1\\right) \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} \\frac{\\partial}{\\partial {\\theta} } - \\frac{1}{r} \\left(r + \\frac{\\sin{\\left (\\phi \\right )}}{\\sin{\\left (\\theta \\right )}}\\right) \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = (r**3*sin(phi)*sin(theta)*cos(theta)**2 + 1)*sin(theta)*cos(phi) d/dr + (-r**3*sin(phi)*sin(theta)**3 + 1)*cos(phi)*cos(theta)/r d/dtheta - (r + sin(phi)/sin(theta))/r d/dphi"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display(Y.frame(), Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial x } = \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\frac{\\partial}{\\partial r } + \\frac{1}{4 r \\sqrt{\\sin^{2}{\\left (\\theta \\right )}}} \\left(- \\sin{\\left (\\phi - 2 \\theta \\right )} + \\sin{\\left (\\phi + 2 \\theta \\right )}\\right) \\frac{\\partial}{\\partial {\\theta} } - \\frac{\\sin{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}} \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "d/dx = sin(theta)*cos(phi) d/dr + (-sin(phi - 2*theta) + sin(phi + 2*theta))/(4*r*sqrt(sin(theta)**2)) d/dtheta - sin(phi)/(r*sin(theta)) d/dphi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial y } = \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\frac{\\partial}{\\partial r } + \\frac{1}{4 r \\sqrt{\\sin^{2}{\\left (\\theta \\right )}}} \\left(\\cos{\\left (\\phi - 2 \\theta \\right )} - \\cos{\\left (\\phi + 2 \\theta \\right )}\\right) \\frac{\\partial}{\\partial {\\theta} } + \\frac{\\cos{\\left (\\phi \\right )}}{r \\sin{\\left (\\theta \\right )}} \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "d/dy = sin(phi)*sin(theta) d/dr + (cos(phi - 2*theta) - cos(phi + 2*theta))/(4*r*sqrt(sin(theta)**2)) d/dtheta + cos(phi)/(r*sin(theta)) d/dphi"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial z } = \\cos{\\left (\\theta \\right )} \\frac{\\partial}{\\partial r } - \\frac{1}{r} \\sin{\\left (\\theta \\right )} \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/plain": [
       "d/dz = cos(theta) d/dr - sin(theta)/r d/dtheta"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in M.irange(): \n",
    "    show(e[i].display(Y.frame(), Y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The components of a tensor field w.r.t. the default frame can also be obtained as a list, via the command <span style=\"font-family: courier new,courier;\">[:]</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[y + 1, - x, x y z\\right]</script></html>"
      ],
      "text/plain": [
       "[y + 1, -x, x*y*z]"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>An alternative is to use the method <span style=\"font-family: courier new,courier;\">display_comp()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} v_{\\phantom{\\, x}}^{ \\, x } & = & y + 1 \\\\ v_{\\phantom{\\, y}}^{ \\, y } & = & - x \\\\ v_{\\phantom{\\, z}}^{ \\, z } & = & x y z \\end{array}</script></html>"
      ],
      "text/plain": [
       "v^x = y + 1 \n",
       "v^y = -x \n",
       "v^z = x*y*z "
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display_comp()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>To obtain the components w.r.t. to another frame, one may go through the method <span style=\"font-family: courier new,courier;\">comp()</span> and specify the frame:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\frac{x \\left(y z^{2} + 1\\right)}{\\sqrt{x^{2} + y^{2} + z^{2}}}, \\frac{x z \\left(- x^{2} y - y^{3} + 1\\right)}{\\sqrt{x^{2} + y^{2}} \\left(x^{2} + y^{2} + z^{2}\\right)}, - \\frac{x^{2} + y^{2} + y}{x^{2} + y^{2}}\\right]</script></html>"
      ],
      "text/plain": [
       "[x*(y*z**2 + 1)/sqrt(x**2 + y**2 + z**2),\n",
       " x*z*(-x**2*y - y**3 + 1)/(sqrt(x**2 + y**2)*(x**2 + y**2 + z**2)),\n",
       " -(x**2 + y**2 + y)/(x**2 + y**2)]"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.comp(Y.frame())[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>However a shortcut is to provide the frame as the first argument of the square brackets:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\frac{x \\left(y z^{2} + 1\\right)}{\\sqrt{x^{2} + y^{2} + z^{2}}}, \\frac{x z \\left(- x^{2} y - y^{3} + 1\\right)}{\\sqrt{x^{2} + y^{2}} \\left(x^{2} + y^{2} + z^{2}\\right)}, - \\frac{x^{2} + y^{2} + y}{x^{2} + y^{2}}\\right]</script></html>"
      ],
      "text/plain": [
       "[x*(y*z**2 + 1)/sqrt(x**2 + y**2 + z**2),\n",
       " x*z*(-x**2*y - y**3 + 1)/(sqrt(x**2 + y**2)*(x**2 + y**2 + z**2)),\n",
       " -(x**2 + y**2 + y)/(x**2 + y**2)]"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[Y.frame(), :]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} v_{\\phantom{\\, r}}^{ \\, r } & = & \\frac{x \\left(y z^{2} + 1\\right)}{\\sqrt{x^{2} + y^{2} + z^{2}}} \\\\ v_{\\phantom{\\, {\\theta}}}^{ \\, {\\theta} } & = & \\frac{x z \\left(- x^{2} y - y^{3} + 1\\right)}{\\sqrt{x^{2} + y^{2}} \\left(x^{2} + y^{2} + z^{2}\\right)} \\\\ v_{\\phantom{\\, {\\phi}}}^{ \\, {\\phi} } & = & - \\frac{x^{2} + y^{2} + y}{x^{2} + y^{2}} \\end{array}</script></html>"
      ],
      "text/plain": [
       "v^r = x*(y*z**2 + 1)/sqrt(x**2 + y**2 + z**2) \n",
       "v^theta = x*z*(-x**2*y - y**3 + 1)/(sqrt(x**2 + y**2)*(x**2 + y**2 + z**2)) \n",
       "v^phi = -(x**2 + y**2 + y)/(x**2 + y**2) "
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display_comp(Y.frame())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Components are shown expressed in terms of the default's coordinates; to get them in terms of the coordinates $(r,\\theta,\\phi)$ instead, add the chart name as the last argument in the square brackets:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(r^{3} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos^{2}{\\left (\\theta \\right )} + 1\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}, \\frac{1}{r} \\left(- r^{3} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + 1\\right) \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )}, - \\frac{1}{r} \\left(r + \\frac{\\sin{\\left (\\phi \\right )}}{\\sin{\\left (\\theta \\right )}}\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[(r**3*sin(phi)*sin(theta)*cos(theta)**2 + 1)*sin(theta)*cos(phi),\n",
       " (-r**3*sin(phi)*sin(theta)**3 + 1)*cos(phi)*cos(theta)/r,\n",
       " -(r + sin(phi)/sin(theta))/r]"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[Y.frame(), :, Y]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>or specify the chart in <span style=\"font-family: courier new,courier;\">display_comp()</span>:<span style=\"font-family: courier new,courier;\"><br /></span></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} v_{\\phantom{\\, r}}^{ \\, r } & = & \\left(r^{3} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos^{2}{\\left (\\theta \\right )} + 1\\right) \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\\\ v_{\\phantom{\\, {\\theta}}}^{ \\, {\\theta} } & = & \\frac{1}{r} \\left(- r^{3} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + 1\\right) \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} \\\\ v_{\\phantom{\\, {\\phi}}}^{ \\, {\\phi} } & = & - \\frac{1}{r} \\left(r + \\frac{\\sin{\\left (\\phi \\right )}}{\\sin{\\left (\\theta \\right )}}\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "v^r = (r**3*sin(phi)*sin(theta)*cos(theta)**2 + 1)*sin(theta)*cos(phi) \n",
       "v^theta = (-r**3*sin(phi)*sin(theta)**3 + 1)*cos(phi)*cos(theta)/r \n",
       "v^phi = -(r + sin(phi)/sin(theta))/r "
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display_comp(Y.frame(), chart=Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To get some vector component as a scalar field instead of a coordinate expression, use double square brackets:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(v[[1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & y + 1 \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + 1 \\end{array}</script></html>"
      ],
      "text/plain": [
       "U --> R\n",
       "(x, y, z) |--> y + 1\n",
       "(r, theta, phi) |--> r*sin(phi)*sin(theta) + 1"
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[[1]].display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAASBAMAAADWL/HSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZIu+JVM27RDKr\nZt2dj8xZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAl0lEQVQYGWNgIAGwB6IoZleAcGVsPmIVZ2DC\nI86orMDgKgDWBzMHrJ6NPYBBH2Icirg7cwPDTCziAqwKDJ+A4mwdHX0RHR0HQErA9q4X4P4N4jAw\noJjDYMHAFIBNfA4D5wFs4pYM6y9gE5dJsndAFWc++uMsSMQCIoywF8TnaWD8BRVndIAyQBTfBU4F\nJC6cyaVkDWcjMQCNESDTdOqp9QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$y + 1$$"
      ],
      "text/plain": [
       "y + 1"
      ]
     },
     "execution_count": 143,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[[1]].expr(X_U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A vector field can be defined with components being unspecified functions of the coordinates:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = \\operatorname{u_{x}}{\\left (x,y,z \\right )} \\frac{\\partial}{\\partial x } + \\operatorname{u_{y}}{\\left (x,y,z \\right )} \\frac{\\partial}{\\partial y } + \\operatorname{u_{z}}{\\left (x,y,z \\right )} \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "u = u_x(x, y, z) d/dx + u_y(x, y, z) d/dy + u_z(x, y, z) d/dz"
      ]
     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = U.vector_field(name='u')\n",
    "u[:] = [function('u_x')(x,y,z), function('u_y')(x,y,z), function('u_z')(x,y,z)]\n",
    "u.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s = \\left( y + \\operatorname{u_{x}}{\\left (x,y,z \\right )} + 1 \\right) \\frac{\\partial}{\\partial x } + \\left( - x + \\operatorname{u_{y}}{\\left (x,y,z \\right )} \\right) \\frac{\\partial}{\\partial y } + \\left( x y z + \\operatorname{u_{z}}{\\left (x,y,z \\right )} \\right) \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "s = (y + u_x(x, y, z) + 1) d/dx + (-x + u_y(x, y, z)) d/dy + (x*y*z + u_z(x, y, z)) d/dz"
      ]
     },
     "execution_count": 145,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = v + u ; s.set_name('s') ; s.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h3>Values of vector fields at a given point</h3>\n",
    "<p>The value of a vector field at some point of the manifold is obtained via the method <span style=\"font-family: courier new,courier;\">at()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangent vector v at Point p on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "vp = v.at(p) ; print(vp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = 3 \\frac{\\partial}{\\partial x } -\\frac{\\partial}{\\partial y } -2 \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "v = 3 d/dx - d/dy - 2 d/dz"
      ]
     },
     "execution_count": 147,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vp.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Indeed, recall that, w.r.t. chart X_U=$(x,y,z)$,  the coordinates of the point $p$ and the components of the vector field $v$ are</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1, 2, -1\\right)</script></html>"
      ],
      "text/plain": [
       "(1, 2, -1)"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.coord(X_U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\left( y + 1 \\right) \\frac{\\partial}{\\partial x } - x \\frac{\\partial}{\\partial y } + x y z \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "v = (y + 1) d/dx - x d/dy + x*y*z d/dz"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display(X_U.frame(), X_U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Note that to simplify the writing, the symbol used to denote the value of the vector field at point $p$ is the same as that of the vector field itself (namely $v$); this can be changed by the method <span style=\"font-family: courier new,courier;\">set_name()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v|_p = 3 \\frac{\\partial}{\\partial x } -\\frac{\\partial}{\\partial y } -2 \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "v = 3 d/dx - d/dy - 2 d/dz"
      ]
     },
     "execution_count": 150,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vp.set_name(latex_name='v|_p')\n",
    "vp.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Of course, $v|_p$ belongs to the tangent space at $p$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T_{p}\\,\\mathcal{M}</script></html>"
      ],
      "text/plain": [
       "Tangent space at Point p on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vp.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 152,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vp in M.tangent_space(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangent vector u at Point p on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "up = u.at(p) ; print(up)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = \\verb|u_x(1,|\\phantom{\\verb!x!}\\verb|2,|\\phantom{\\verb!x!}\\verb|-1)| \\frac{\\partial}{\\partial x } + \\verb|u_y(1,|\\phantom{\\verb!x!}\\verb|2,|\\phantom{\\verb!x!}\\verb|-1)| \\frac{\\partial}{\\partial y } + \\verb|u_z(1,|\\phantom{\\verb!x!}\\verb|2,|\\phantom{\\verb!x!}\\verb|-1)| \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "u = u_x(1, 2, -1) d/dx + u_y(1, 2, -1) d/dy + u_z(1, 2, -1) d/dz"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "up.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>1-forms</h2>\n",
    "<p>A 1-form on $\\mathcal{M}$ is a field of linear forms. For instance, it can be the <strong>differential of a scalar field</strong>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1-form df on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "df = f.differential() ; print(df)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}f = \\mathrm{d} x + 2 y \\mathrm{d} y + 3 z^{2} \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "df = dx + 2*y dy + 3*z**2 dz"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In the above writing, the 1-form is expanded over the basis $(\\mathrm{d}x, \\mathrm{d}y, \\mathrm{d}z)$ associated with the chart $(x,y,z)$. This basis can be accessed via the method <span style=\"font-family: courier new,courier;\">coframe()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathcal{M}, \\left(\\mathrm{d} x,\\mathrm{d} y,\\mathrm{d} z\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "Coordinate coframe (M, (dx,dy,dz))"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dX = X.coframe() ; dX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The list of all coframes defined on a given manifold open subset is returned by the method <span style=\"font-family: courier new,courier;\">coframes()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathcal{M}, \\left(\\mathrm{d} x,\\mathrm{d} y,\\mathrm{d} z\\right)\\right), \\left(U, \\left(\\mathrm{d} x,\\mathrm{d} y,\\mathrm{d} z\\right)\\right), \\left(U, \\left(\\mathrm{d} r,\\mathrm{d} {\\theta},\\mathrm{d} {\\phi}\\right)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Coordinate coframe (M, (dx,dy,dz)),\n",
       " Coordinate coframe (U, (dx,dy,dz)),\n",
       " Coordinate coframe (U, (dr,dtheta,dphi))]"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.coframes()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>As for a vector field, the value of the differential form at some point on the manifold is obtained by the method <span style=\"font-family: courier new,courier;\">at()</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Linear form df on the Tangent space at Point p on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "dfp = df.at(p) ; print(dfp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}f = \\mathrm{d} x + 4 \\mathrm{d} y + 3 \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "df = dx + 4 dy + 3 dz"
      ]
     },
     "execution_count": 160,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfp.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Recall that</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1, 2, -1\\right)</script></html>"
      ],
      "text/plain": [
       "(1, 2, -1)"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p.coord()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The linear form $\\mathrm{d}f|_p$ belongs to the dual of the tangent vector space at $p$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T_{p}\\,\\mathcal{M}^*</script></html>"
      ],
      "text/plain": [
       "Dual of the Tangent space at Point p on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfp.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfp.parent() is M.tangent_space(p).dual()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>As such, it is acting on vectors at $p$, yielding a real number:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangent vector v at Point p on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v|_p = 3 \\frac{\\partial}{\\partial x } -\\frac{\\partial}{\\partial y } -2 \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "v = 3 d/dx - d/dy - 2 d/dz"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(vp) ; vp.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMquZiWYiRHa7\nVO+0fdwUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVUlEQVQIHWNggALnsPLyAhiHof////8BcF4m\nAwMrnMMAVFWC4DEwcG1A5vEgcxgqkHmMH4A8IWMQUGFg4FZAlmM+gMyTR9gNFPZPQJa7j8Kbd4GB\nAQCGAQ9DQk+A3gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$-7$$"
      ],
      "text/plain": [
       "-7"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfp(vp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tangent vector u at Point p on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}u = \\verb|u_x(1,|\\phantom{\\verb!x!}\\verb|2,|\\phantom{\\verb!x!}\\verb|-1)| \\frac{\\partial}{\\partial x } + \\verb|u_y(1,|\\phantom{\\verb!x!}\\verb|2,|\\phantom{\\verb!x!}\\verb|-1)| \\frac{\\partial}{\\partial y } + \\verb|u_z(1,|\\phantom{\\verb!x!}\\verb|2,|\\phantom{\\verb!x!}\\verb|-1)| \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "u = u_x(1, 2, -1) d/dx + u_y(1, 2, -1) d/dy + u_z(1, 2, -1) d/dz"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(up) ; up.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYsAAAAVBAMAAABF1d8/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIkQyEJm7VHbNq2bv\nid2NdxwmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADgUlEQVRYCa2XTUhUURTH/2+ccZ7OhwPhSpSJ\naNHGZtPaiQhSDGeRFIJkEba1RRFROkGbVklBOC6yoCI0clq4CFxEhBBEuTEMsiaQIFBTETWo7Nzz\n7n1fM/qGN3MWc889797f/P/vvvs+AGc0ObslepFEiWJxqVogbw5eFv271mcrad39KVtXpY9UYrVH\nrFRmvkDNCxfdoHI49UUnto4q4beS1YHYugt7PQ20u2rU/VFU8gPSMhjKu0heHCE2OOyahCmgo2dD\nVh8D7x0Dgk8m00AsK4tX1UF9RmVm6wekb6GhYBKMZG+OFDvvmoTvVNCVjQ/AZMo54k6azL+WNdNG\n033nKOr5AQXeoMt9Yr04LLYT0A7g2H6lQitQZtoYS5e0gbtyuGljVtioBogvKjvIUxCLrUuicRNX\ntqUu6Bm7Dcp/pdUhoxWrgTlZUzYiCWGjCiAEl10gT0FsI0aySYJpozZPGHM1gMBfqVg1bOOV7Ckb\nJwSjGiDt6ykXyFMQi43QpdhvsxEfIIzNRvyhwNqCbQzKgrIxa9ioHAR8SRLbBvIUxGLDBcckRAXF\nZqObuhTBBysUP1MA22gTtc+53FIuN0pZOFVswxeIWFFxs7fZ2IWDdiFo5Z4UW7PqmOS2wVcmca2w\nbFBNrsZJvp6qANISqN0i7u42igXxOTdtHNqKbRLAtYaXQWRHsI1WWZI2ro2P/3lGJf731vXgO5rj\nC9SwYbPRdebbQBkcthEoAE+hbQMjoRQpce6omgz0UjbcWxyYEcYYpG0g4xcU7UN8TYGaxWnxFsQ2\nxBafRngmjc7TNAnizsV7gx9DLQtzL1wPJF6NXjGUQm1xQKykBI1oecp9gSJ5DGUVaJ8utoknh22E\nZmnk/K2lRUT40S1WJ/T89wTii8QY29n5Z2TU4ej5OEoiaWdxmDY+7UxQwQAN6uKYP9DB7vNisgFq\nS6a9OSwWdSkxTcTx29yIZz/HJZXAymRJK8jEtGGOpSR6g3uVg5ZB55hfahhoybAyPkA/LSrBhdqM\nyOlNzAiBcGeyYL4aHlVD7K2+xr2KQfXDgSyRyuFgXgmYXg1tipWpTRiVYNZo6ZlhZqpS4kVdHaI2\n2MedikGHe88OEKkcTvGLuvpKibAW8WNlqnROJSXbmiSXKwcZ9HI4yqpNj/c3494fsVMxCasUpDR5\nc0p8xKrJvtvGm76nVjLxP5tdbciPDDcZAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\operatorname{u_{x}}{\\left (1,2,-1 \\right )} + 4 \\operatorname{u_{y}}{\\left (1,2,-1 \\right )} + 3 \\operatorname{u_{z}}{\\left (1,2,-1 \\right )}$$"
      ],
      "text/plain": [
       "u_x(1, 2, -1) + 4*u_y(1, 2, -1) + 3*u_z(1, 2, -1)"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfp(up)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The differential 1-form of the unspecified scalar field $h$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} h:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & H{\\left (x,y,z \\right )} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & H{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "h: U --> R\n",
       "   (x, y, z) |--> H(x, y, z)\n",
       "   (r, theta, phi) |--> H(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}h = \\frac{\\partial}{\\partial x} H{\\left (x,y,z \\right )} \\mathrm{d} x + \\frac{\\partial}{\\partial y} H{\\left (x,y,z \\right )} \\mathrm{d} y + \\frac{\\partial}{\\partial z} H{\\left (x,y,z \\right )} \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "dh = Derivative(H(x, y, z), x) dx + Derivative(H(x, y, z), y) dy + Derivative(H(x, y, z), z) dz"
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h.display() ; dh = h.differential() ; dh.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A 1-form <span id=\"cell_outer_132\">can also be defined from scratch:</span></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1-form omega on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "om = U.one_form('omega', r'\\omega') ; print(om)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>It can be specified by providing its components in a given coframe:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = \\left( x^{2} + y^{2} \\right) \\mathrm{d} x + z \\mathrm{d} y + \\left( x - z \\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "omega = (x**2 + y**2) dx + z dy + (x - z) dz"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "om[:] = [x^2+y^2, z, x-z]    # components in the default coframe (dx,dy,dz)\n",
    "om.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Of course, one may set the components in a frame different from the default one:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\mathrm{d} r + r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "omega = r*sin(theta)*cos(phi) dr + r*sin(phi)*sin(theta) dphi"
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "om[Y.frame(), :, Y] = [r*sin(th)*cos(ph), 0, r*sin(th)*sin(ph)]\n",
    "om.display(Y.frame(), Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The components in the coframe $(\\mathrm{d}x,\\mathrm{d}y,\\mathrm{d}z)$ are updated automatically:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = \\left( \\frac{x^{4} + x^{2} y^{2} - y^{2} \\sqrt{x^{2} + y^{2} + z^{2}}}{\\left(x^{2} + y^{2}\\right) \\sqrt{x^{2} + y^{2} + z^{2}}} \\right) \\mathrm{d} x + \\left( \\frac{x y \\left(x^{2} + y^{2} + \\sqrt{x^{2} + y^{2} + z^{2}}\\right)}{\\left(x^{2} + y^{2}\\right) \\sqrt{x^{2} + y^{2} + z^{2}}} \\right) \\mathrm{d} y + \\left( \\frac{x z}{\\sqrt{x^{2} + y^{2} + z^{2}}} \\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "omega = (x**4 + x**2*y**2 - y**2*sqrt(x**2 + y**2 + z**2))/((x**2 + y**2)*sqrt(x**2 + y**2 + z**2)) dx + x*y*(x**2 + y**2 + sqrt(x**2 + y**2 + z**2))/((x**2 + y**2)*sqrt(x**2 + y**2 + z**2)) dy + x*z/sqrt(x**2 + y**2 + z**2) dz"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "om.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let us revert to the values set previously:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = \\left( x^{2} + y^{2} \\right) \\mathrm{d} x + z \\mathrm{d} y + \\left( x - z \\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "omega = (x**2 + y**2) dx + z dy + (x - z) dz"
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "om[:] = [x^2+y^2, z, x-z]\n",
    "om.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>This time, the components in the coframe $(\\mathrm{d}r, \\mathrm{d}\\theta,\\mathrm{d}\\phi)$ are those that are updated:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = r \\left(r \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\frac{\\sqrt{2}}{4} \\left(\\cos{\\left (\\phi - 2 \\theta + \\frac{\\pi}{4} \\right )} - \\cos{\\left (\\phi + 2 \\theta + \\frac{\\pi}{4} \\right )}\\right) - \\cos^{2}{\\left (\\theta \\right )}\\right) \\mathrm{d} r + r^{2} \\left(r \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + \\sin{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} - \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\frac{1}{2} \\sin{\\left (2 \\theta \\right )}\\right) \\mathrm{d} {\\theta} + \\frac{r^{2}}{4} \\left(- 4 r \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} - \\sin{\\left (\\phi - 2 \\theta \\right )} + \\sin{\\left (\\phi + 2 \\theta \\right )}\\right) \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "omega = r*(r*sin(theta)**3*cos(phi) + sqrt(2)*(cos(phi - 2*theta + pi/4) - cos(phi + 2*theta + pi/4))/4 - cos(theta)**2) dr + r**2*(r*sin(theta)**2*cos(phi)*cos(theta) + sin(phi)*cos(theta)**2 - sin(theta)**2*cos(phi) + sin(2*theta)/2) dtheta + r**2*(-4*r*sin(phi)*sin(theta)**3 - sin(phi - 2*theta) + sin(phi + 2*theta))/4 dphi"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "om.display(Y.frame(), Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A 1-form acts on vector fields, resulting in a scalar field:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field omega(v) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\omega\\left(v\\right):& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x^{2} y z + x^{2} y + x^{2} - x y z^{2} - x z + y^{3} + y^{2} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{2} \\left(- \\frac{r^{2}}{4} \\left(\\cos{\\left (2 \\phi - \\theta \\right )} - \\cos{\\left (2 \\phi + \\theta \\right )}\\right) - r^{2} \\sin^{3}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + r^{2} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + r^{2} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + r \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} - \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )}\\right) \\sin{\\left (\\theta \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "omega(v): U --> R\n",
       "   (x, y, z) |--> x**2*y*z + x**2*y + x**2 - x*y*z**2 - x*z + y**3 + y**2\n",
       "   (r, theta, phi) |--> r**2*(-r**2*(cos(2*phi - theta) - cos(2*phi + theta))/4 - r**2*sin(phi)**3*sin(theta)**2*cos(theta) + r**2*sin(phi)*sin(theta)**3*cos(phi) + r**2*sin(phi)*sin(theta)**2*cos(theta) + r*sin(phi)*sin(theta)**2 + sin(theta) - cos(phi)*cos(theta))*sin(theta)"
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display() ; om.display() ; print(om(v)) ; om(v).display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field df(v) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{d}f\\left(v\\right):& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & 3 x y z^{3} - 2 x y + y + 1 \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & 3 r^{5} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} - 2 r^{2} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + 1 \\end{array}</script></html>"
      ],
      "text/plain": [
       "df(v): U --> R\n",
       "   (x, y, z) |--> 3*x*y*z**3 - 2*x*y + y + 1\n",
       "   (r, theta, phi) |--> 3*r**5*sin(phi)*sin(theta)**2*cos(phi)*cos(theta)**3 - 2*r**2*sin(phi)*sin(theta)**2*cos(phi) + r*sin(phi)*sin(theta) + 1"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.display() ; print(df(v)) ; df(v).display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\omega\\left(u\\right):& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x^{2} \\operatorname{u_{x}}{\\left (x,y,z \\right )} + x \\operatorname{u_{z}}{\\left (x,y,z \\right )} + y^{2} \\operatorname{u_{x}}{\\left (x,y,z \\right )} + z \\operatorname{u_{y}}{\\left (x,y,z \\right )} - z \\operatorname{u_{z}}{\\left (x,y,z \\right )} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\left(r \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\cos{\\left (\\theta \\right )} + \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\cos{\\left (\\theta \\right )}\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "omega(u): U --> R\n",
       "   (x, y, z) |--> x**2*u_x(x, y, z) + x*u_z(x, y, z) + y**2*u_x(x, y, z) + z*u_y(x, y, z) - z*u_z(x, y, z)\n",
       "   (r, theta, phi) |--> r*(r*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)**2 + u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*cos(theta) + u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)*cos(phi) - u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*cos(theta))"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.display() ; om(u).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In the case of a differential 1-form, the following identity holds:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df(v) == v(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>1-forms are Sage <em>element</em> objects, whose <em>parent</em> is the $C^\\infty(U)$-module $\\Omega^{1}(U)$ of all 1-forms defined on $U$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Omega^{1}\\left(U\\right)</script></html>"
      ],
      "text/plain": [
       "Free module Omega^1(U) of 1-forms on the Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Free module Omega^1(U) of 1-forms on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(df.parent())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Free module Omega^1(U) of 1-forms on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(om.parent())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>$\\Omega^{1}(U)$ is actually the dual of the free module $\\mathfrak{X}(U)$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.parent() is v.parent().dual()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Differential forms and exterior calculus</h2>\n",
    "<p>The <strong>exterior product</strong> of two 1-forms is taken via the method <span style=\"font-family: courier new,courier;\">wedge()</span> and results in a 2-form:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2-form omega/\\df on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega\\wedge \\mathrm{d}f = \\left( 2 x^{2} y + 2 y^{3} - z \\right) \\mathrm{d} x\\wedge \\mathrm{d} y + \\left( 3 x^{2} z^{2} - x + 3 y^{2} z^{2} + z \\right) \\mathrm{d} x\\wedge \\mathrm{d} z + \\left( - 2 x y + 2 y z + 3 z^{3} \\right) \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "omega/\\df = (2*x**2*y + 2*y**3 - z) dx/\\dy + (3*x**2*z**2 - x + 3*y**2*z**2 + z) dx/\\dz + (-2*x*y + 2*y*z + 3*z**3) dy/\\dz"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = om.wedge(df) ; print(a) ; a.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A matrix view of the components:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrr}\n",
       "0 & 2 \\, x^{2} y + 2 \\, y^{3} - z & 3 \\, x^{2} z^{2} + 3 \\, y^{2} z^{2} - x + z \\\\\n",
       "-2 \\, x^{2} y - 2 \\, y^{3} + z & 0 & 3 \\, z^{3} - 2 \\, x y + 2 \\, y z \\\\\n",
       "-3 \\, x^{2} z^{2} - 3 \\, y^{2} z^{2} + x - z & -3 \\, z^{3} + 2 \\, x y - 2 \\, y z & 0\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[                             0            2*x^2*y + 2*y^3 - z  3*x^2*z^2 + 3*y^2*z^2 - x + z]\n",
       "[          -2*x^2*y - 2*y^3 + z                              0          3*z^3 - 2*x*y + 2*y*z]\n",
       "[-3*x^2*z^2 - 3*y^2*z^2 + x - z         -3*z^3 + 2*x*y - 2*y*z                              0]"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Displaying only the non-vanishing components, skipping the redundant ones (i.e. those that can be deduced by antisymmetry):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\omega\\wedge \\mathrm{d}f_{ \\, x \\, y }^{ \\phantom{\\, x}\\phantom{\\, y} } & = & 2 x^{2} y + 2 y^{3} - z \\\\ \\omega\\wedge \\mathrm{d}f_{ \\, x \\, z }^{ \\phantom{\\, x}\\phantom{\\, z} } & = & 3 x^{2} z^{2} - x + 3 y^{2} z^{2} + z \\\\ \\omega\\wedge \\mathrm{d}f_{ \\, y \\, z }^{ \\phantom{\\, y}\\phantom{\\, z} } & = & - 2 x y + 2 y z + 3 z^{3} \\end{array}</script></html>"
      ],
      "text/plain": [
       "omega/\\df_xy = 2*x**2*y + 2*y**3 - z \n",
       "omega/\\df_xz = 3*x**2*z**2 - x + 3*y**2*z**2 + z \n",
       "omega/\\df_yz = -2*x*y + 2*y*z + 3*z**3 "
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.display_comp(only_nonredundant=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The 2-form $\\omega\\wedge\\mathrm{d}f$ can be expanded on the $(\\mathrm{d}r,\\mathrm{d}\\theta,\\mathrm{d}\\phi)$ coframe:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega\\wedge \\mathrm{d}f = r^{2} \\left(3 r^{3} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - 3 r^{3} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - 3 r^{2} \\sin{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} - 2 r \\sin^{2}{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - 2 r \\sin^{2}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\phi \\right )} + 2 r \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\sin^{2}{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} - \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )}\\right) \\mathrm{d} r\\wedge \\mathrm{d} {\\theta} + r^{2} \\left(3 r^{3} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\theta \\right )} + 2 r^{2} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} - 3 r^{2} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 6 r^{2} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - 3 r^{2} \\cos{\\left (\\phi \\right )} - \\frac{r}{2} \\left(\\cos{\\left (2 \\phi - \\theta \\right )} - \\cos{\\left (2 \\phi + \\theta \\right )}\\right) - 2 r \\sin^{3}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + 2 r \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 2 r \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\phi \\right )} - \\frac{1}{2} \\sin{\\left (2 \\theta \\right )} - \\frac{1}{8} \\cos{\\left (2 \\left(\\phi - \\theta\\right) \\right )} + \\frac{1}{8} \\cos{\\left (2 \\left(\\phi + \\theta\\right) \\right )}\\right) \\sin{\\left (\\theta \\right )} \\mathrm{d} r\\wedge \\mathrm{d} {\\phi} + r^{3} \\left(3 r^{3} \\sin{\\left (\\phi \\right )} \\sin^{5}{\\left (\\theta \\right )} - 3 r^{3} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + 2 r^{2} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + 3 r^{2} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} + 2 r \\sin^{3}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} - 2 r \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + 2 r \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\sin^{2}{\\left (\\theta \\right )} - \\frac{1}{4} \\cos{\\left (\\phi - 2 \\theta \\right )} + \\frac{1}{4} \\cos{\\left (\\phi + 2 \\theta \\right )} - 1\\right) \\sin{\\left (\\theta \\right )} \\mathrm{d} {\\theta}\\wedge \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "omega/\\df = r**2*(3*r**3*sin(theta)**4*cos(phi) - 3*r**3*sin(theta)**2*cos(phi) - 3*r**2*sin(phi)*cos(theta)**3 - 2*r*sin(phi)**2*sin(theta)*cos(theta) - 2*r*sin(theta)**2*cos(phi)**3 + 2*r*sin(theta)**2*cos(phi) - sin(phi)**2*sin(theta) + sin(theta) - cos(phi)*cos(theta)) dr/\\dtheta + r**2*(3*r**3*sin(phi)*sin(theta)**2*cos(theta)**3 + 2*r**2*sin(phi)*sin(theta)**4 - 3*r**2*sin(theta)**4*cos(phi) + 6*r**2*sin(theta)**2*cos(phi) - 3*r**2*cos(phi) - r*(cos(2*phi - theta) - cos(2*phi + theta))/2 - 2*r*sin(phi)**3*sin(theta)**2*cos(theta) + 2*r*sin(phi)*sin(theta)**3*cos(phi) + 2*r*sin(phi)*sin(theta)**2*cos(theta) - sin(phi)*sin(theta)**2 + sin(phi) - sin(2*theta)/2 - cos(2*(phi - theta))/8 + cos(2*(phi + theta))/8)*sin(theta) dr/\\dphi + r**3*(3*r**3*sin(phi)*sin(theta)**5 - 3*r**3*sin(phi)*sin(theta)**3 + 2*r**2*sin(phi)*sin(theta)**3*cos(theta) + 3*r**2*sin(theta)*cos(phi)*cos(theta)**3 + 2*r*sin(phi)**3*sin(theta)**3 - 2*r*sin(phi)*sin(theta)**3 + 2*r*sin(phi)*sin(theta)**2*cos(phi)*cos(theta) + sin(phi)*sin(theta)**2*cos(phi) + sin(theta)**2 - cos(phi - 2*theta)/4 + cos(phi + 2*theta)/4 - 1)*sin(theta) dtheta/\\dphi"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.display(Y.frame(), Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>As a 2-form, $A:=\\omega\\wedge\\mathrm{d}f$ can be applied to a pair of vectors and is antisymmetric:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field A(u,v) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} A\\left(u,v\\right):& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & 3 x^{3} y z^{3} \\operatorname{u_{x}}{\\left (x,y,z \\right )} - 2 x^{3} y \\operatorname{u_{x}}{\\left (x,y,z \\right )} - 2 x^{2} y^{2} z \\operatorname{u_{y}}{\\left (x,y,z \\right )} - 2 x^{2} y^{2} \\operatorname{u_{y}}{\\left (x,y,z \\right )} - 3 x^{2} y z^{2} \\operatorname{u_{z}}{\\left (x,y,z \\right )} - x^{2} y z \\operatorname{u_{x}}{\\left (x,y,z \\right )} - 2 x^{2} y \\operatorname{u_{y}}{\\left (x,y,z \\right )} - 2 x^{2} y \\operatorname{u_{z}}{\\left (x,y,z \\right )} - 3 x^{2} z^{2} \\operatorname{u_{z}}{\\left (x,y,z \\right )} + 3 x y^{3} z^{3} \\operatorname{u_{x}}{\\left (x,y,z \\right )} - 2 x y^{3} \\operatorname{u_{x}}{\\left (x,y,z \\right )} + 2 x y^{2} z^{2} \\operatorname{u_{y}}{\\left (x,y,z \\right )} + 3 x y z^{4} \\operatorname{u_{y}}{\\left (x,y,z \\right )} + x y z^{2} \\operatorname{u_{x}}{\\left (x,y,z \\right )} + 2 x y z \\operatorname{u_{z}}{\\left (x,y,z \\right )} + x y \\operatorname{u_{z}}{\\left (x,y,z \\right )} + 3 x z^{3} \\operatorname{u_{z}}{\\left (x,y,z \\right )} + x z \\operatorname{u_{x}}{\\left (x,y,z \\right )} + x \\operatorname{u_{z}}{\\left (x,y,z \\right )} - 2 y^{4} \\operatorname{u_{y}}{\\left (x,y,z \\right )} - 3 y^{3} z^{2} \\operatorname{u_{z}}{\\left (x,y,z \\right )} - 2 y^{3} \\operatorname{u_{y}}{\\left (x,y,z \\right )} - 3 y^{2} z^{2} \\operatorname{u_{z}}{\\left (x,y,z \\right )} + y z \\operatorname{u_{y}}{\\left (x,y,z \\right )} - y z \\operatorname{u_{z}}{\\left (x,y,z \\right )} + z \\operatorname{u_{y}}{\\left (x,y,z \\right )} - z \\operatorname{u_{z}}{\\left (x,y,z \\right )} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\left(3 r^{6} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{3}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} + 3 r^{6} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} + 3 r^{5} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{6}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - 6 r^{5} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 3 r^{5} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 2 r^{4} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{4}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - 2 r^{4} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{2}{\\left (\\phi \\right )} \\sin^{5}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - 2 r^{4} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{2}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + 2 r^{4} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{2}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 3 r^{4} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{5}{\\left (\\theta \\right )} - 3 r^{4} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} - 2 r^{3} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{3}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + r^{3} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{3}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - 2 r^{3} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\phi \\right )} - r^{3} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - r^{3} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + r^{3} \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - 2 r^{3} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{2}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} + 3 r^{3} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{4}{\\left (\\theta \\right )} - 3 r^{3} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{2}{\\left (\\theta \\right )} + 3 r^{3} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} - 2 r^{2} \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + 2 r^{2} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{3}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} - 2 r^{2} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + 2 r^{2} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + r \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + r \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + r \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - r \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\cos{\\left (\\theta \\right )} + \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\cos{\\left (\\theta \\right )}\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "A(u,v): U --> R\n",
       "   (x, y, z) |--> 3*x**3*y*z**3*u_x(x, y, z) - 2*x**3*y*u_x(x, y, z) - 2*x**2*y**2*z*u_y(x, y, z) - 2*x**2*y**2*u_y(x, y, z) - 3*x**2*y*z**2*u_z(x, y, z) - x**2*y*z*u_x(x, y, z) - 2*x**2*y*u_y(x, y, z) - 2*x**2*y*u_z(x, y, z) - 3*x**2*z**2*u_z(x, y, z) + 3*x*y**3*z**3*u_x(x, y, z) - 2*x*y**3*u_x(x, y, z) + 2*x*y**2*z**2*u_y(x, y, z) + 3*x*y*z**4*u_y(x, y, z) + x*y*z**2*u_x(x, y, z) + 2*x*y*z*u_z(x, y, z) + x*y*u_z(x, y, z) + 3*x*z**3*u_z(x, y, z) + x*z*u_x(x, y, z) + x*u_z(x, y, z) - 2*y**4*u_y(x, y, z) - 3*y**3*z**2*u_z(x, y, z) - 2*y**3*u_y(x, y, z) - 3*y**2*z**2*u_z(x, y, z) + y*z*u_y(x, y, z) - y*z*u_z(x, y, z) + z*u_y(x, y, z) - z*u_z(x, y, z)\n",
       "   (r, theta, phi) |--> r*(3*r**6*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**3*sin(theta)**4*cos(phi)*cos(theta)**3 + 3*r**6*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**4*cos(phi)**3*cos(theta)**3 + 3*r**5*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**6*cos(phi) - 6*r**5*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**4*cos(phi) + 3*r**5*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**2*cos(phi) + 2*r**4*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**4*sin(theta)**4*cos(theta) - 2*r**4*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**2*sin(theta)**5*cos(phi) - 2*r**4*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**2*sin(theta)**4*cos(theta) + 2*r**4*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**2*sin(theta)**3*cos(phi) + 3*r**4*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**5 - 3*r**4*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**3 - 2*r**3*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**3*sin(theta)**4*cos(phi) + r**3*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**3*sin(theta)**3*cos(theta) - 2*r**3*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**4*cos(phi)**3 - r**3*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**4*cos(phi) - r**3*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**3*cos(theta) + r**3*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**2*cos(phi) - 2*r**3*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**2*sin(theta)**4 + 3*r**3*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)**4 - 3*r**3*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)**2 + 3*r**3*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)*cos(phi)*cos(theta)**3 - 2*r**2*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**3 + 2*r**2*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)**3*sin(theta)**3 - 2*r**2*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**3 + 2*r**2*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**2*cos(phi)*cos(theta) + r*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)*cos(phi)*cos(theta) + r*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)*cos(theta) + r*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**2*cos(phi) - r*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)*cos(theta) + u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*cos(theta) + u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)*cos(phi) - u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*cos(theta))"
      ]
     },
     "execution_count": 188,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.set_name('A')\n",
    "print(a(u,v)) ; a(u,v).display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 189,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(u,v) == - a(v,u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "no symmetry; antisymmetry: (0, 1)\n"
     ]
    }
   ],
   "source": [
    "a.symmetries()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The <strong>exterior derivative</strong>  of a differential form:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2-form domega on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}\\omega = - 2 y \\mathrm{d} x\\wedge \\mathrm{d} y +\\mathrm{d} x\\wedge \\mathrm{d} z -\\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "domega = -2*y dx/\\dy + dx/\\dz - dy/\\dz"
      ]
     },
     "execution_count": 191,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dom = om.exterior_derivative() ; print(dom) ; dom.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Instead of invoking the method <span style=\"font-family: courier new,courier;\">exterior_derivative()</span>, one can use the function <span style=\"font-family: courier new,courier;\">xder</span>, after having imported it from <span style=\"font-family: courier new,courier;\">sage.manifolds.utilities</span>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from sage.manifolds.utilities import xder\n",
    "dom = xder(om)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3-form dA on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}A = \\left( - 6 y z^{2} - 2 y - 1 \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "dA = (-6*y*z**2 - 2*y - 1) dx/\\dy/\\dz"
      ]
     },
     "execution_count": 193,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "da = xder(a) ; print(da) ; da.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The exterior derivative is nilpotent:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}\\mathrm{d}f = 0</script></html>"
      ],
      "text/plain": [
       "ddf = 0"
      ]
     },
     "execution_count": 194,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ddf = xder(df) ; ddf.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}\\mathrm{d}\\omega = 0</script></html>"
      ],
      "text/plain": [
       "ddomega = 0"
      ]
     },
     "execution_count": 195,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ddom = xder(dom) ; ddom.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Lie derivative</h2>\n",
    "<p>The Lie derivative of any tensor field with respect to a vector field is computed by the method <span style=\"font-family: courier new,courier;\">lie_derivative()</span>, with the vector field as the argument:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1-form on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x y z + 2 x - y z^{2} - z \\right) \\mathrm{d} x + \\left( x^{2} z + x^{2} + x y z - x z^{2} + y^{2} \\right) \\mathrm{d} y + \\left( x^{2} y - 2 x y z + y + 1 \\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "(x*y*z + 2*x - y*z**2 - z) dx + (x**2*z + x**2 + x*y*z - x*z**2 + y**2) dy + (x**2*y - 2*x*y*z + y + 1) dz"
      ]
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lv_om = om.lie_derivative(v) ; print(lv_om) ; lv_om.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# lu_dh = dh.lie_derivative(u) ; print(lu_dh) ; lu_dh.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let us check <strong>Cartan identity</strong> on the 1-form $\\omega$:</p>\n",
    "<p>$\\mathcal{L}_v \\omega = v\\cdot \\mathrm{d}\\omega + \\mathrm{d}\\langle \\omega, v\\rangle$</p>\n",
    "<p>and on the 2-form $A$:</p>\n",
    "<p>$\\mathcal{L}_v A = v\\cdot \\mathrm{d}A + \\mathrm{d}(v\\cdot A)$</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 198,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "om.lie_derivative(v) == v.contract(xder(om)) + xder(om(v))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 199,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.lie_derivative(v) == v.contract(xder(a)) + xder(v.contract(a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The Lie derivative of a vector field along another one is the <strong>commutator</strong> of the two vectors fields:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# v.lie_derivative(u)(f) == u(v(f)) - v(u(f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Tensor fields of arbitrary rank</h2>\n",
    "<p>Up to now, we have encountered tensor fields</p>\n",
    "<ul>\n",
    "<li>of type (0,0) (i.e. scalar fields),</li>\n",
    "<li>of type (1,0) (i.e. vector fields),</li>\n",
    "<li>of type (0,1) (i.e. 1-forms),</li>\n",
    "<li>of type (0,2) and antisymmetric (i.e. 2-forms).</li>\n",
    "</ul>\n",
    "<p>More generally, tensor fields of any type $(p,q)$ can be introduced in SageMath. For instance a tensor field of type (1,2) on the open subset $U$ is declared as follows:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field T of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "t = U.tensor_field(1, 2, name='T') ; print(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>As for vectors or 1-forms, the tensor's components with respect to the domain's default frame are set by means of square brackets:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "t[1,2,1] = 1 + x^2\n",
    "t[3,2,1] = x*y*z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Unset components are zero:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T = \\left( x^{2} + 1 \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x + x y z \\frac{\\partial}{\\partial z }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x</script></html>"
      ],
      "text/plain": [
       "T = (x**2 + 1) d/dx*dy*dx + x*y*z d/dz*dy*dx"
      ]
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[0, 0, 0\\right], \\left[x^{2} + 1, 0, 0\\right], \\left[0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0\\right], \\left[0, 0, 0\\right], \\left[0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0\\right], \\left[x y z, 0, 0\\right], \\left[0, 0, 0\\right]\\right]\\right]</script></html>"
      ],
      "text/plain": [
       "[[[0, 0, 0], [x**2 + 1, 0, 0], [0, 0, 0]],\n",
       " [[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       " [[0, 0, 0], [x*y*z, 0, 0], [0, 0, 0]]]"
      ]
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Display of the nonzero components:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} T_{ \\phantom{\\, x} \\, y \\, x }^{ \\, x \\phantom{\\, y} \\phantom{\\, x} } & = & x^{2} + 1 \\\\ T_{ \\phantom{\\, z} \\, y \\, x }^{ \\, z \\phantom{\\, y} \\phantom{\\, x} } & = & x y z \\end{array}</script></html>"
      ],
      "text/plain": [
       "T^x_yx = x**2 + 1 \n",
       "T^z_yx = x*y*z "
      ]
     },
     "execution_count": 205,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.display_comp()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Double square brackets return the component (still w.r.t. the default frame) as a scalar field, while single square brackets return the expression of this scalar field in terms of the domain's default coordinates:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x^{2} + 1 \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{2} \\sin^{2}{\\left (\\theta \\right )} \\cos^{2}{\\left (\\phi \\right )} + 1 \\end{array}</script></html>"
      ],
      "text/plain": [
       "U --> R\n",
       "(x, y, z) |--> x**2 + 1\n",
       "(r, theta, phi) |--> r**2*sin(theta)**2*cos(phi)**2 + 1"
      ]
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(t[[1,2,1]]) ; t[[1,2,1]].display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x**2 + 1\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{2} + 1</script></html>"
      ],
      "text/plain": [
       "x**2 + 1"
      ]
     },
     "execution_count": 207,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(t[1,2,1]) ; t[1,2,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A tensor field of type (1,2) maps a 3-tuple (1-form, vector field, vector field) to a scalar field:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field T(omega,u,v) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} T\\left(\\omega,u,v\\right):& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & \\left(x^{4} y + x^{4} + x^{2} y^{3} + x^{2} y^{2} z + x^{2} y^{2} + x^{2} y z + x^{2} y + x^{2} - x y^{2} z^{2} - x y z^{2} + y^{3} + y^{2}\\right) \\operatorname{u_{y}}{\\left (x,y,z \\right )} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{2} \\left(- r^{3} \\sin^{4}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + \\sqrt{2} r^{3} \\sin^{2}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi + \\frac{\\pi}{4} \\right )} + r^{3} \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - r^{3} \\sin^{2}{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + r^{3} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} + \\frac{r^{2}}{4} \\left(\\cos{\\left (\\phi - 2 \\theta \\right )} - \\cos{\\left (\\phi + 2 \\theta \\right )}\\right) - r^{2} \\sin^{3}{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + \\sqrt{2} r^{2} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi + \\frac{\\pi}{4} \\right )} - \\frac{r^{2}}{2} \\sin{\\left (2 \\phi \\right )} + r^{2} \\sin^{2}{\\left (\\theta \\right )} + r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + 1\\right) \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin^{2}{\\left (\\theta \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "T(omega,u,v): U --> R\n",
       "   (x, y, z) |--> (x**4*y + x**4 + x**2*y**3 + x**2*y**2*z + x**2*y**2 + x**2*y*z + x**2*y + x**2 - x*y**2*z**2 - x*y*z**2 + y**3 + y**2)*u_y(x, y, z)\n",
       "   (r, theta, phi) |--> r**2*(-r**3*sin(phi)**4*sin(theta)**2*cos(theta) + sqrt(2)*r**3*sin(phi)**2*sin(theta)**3*cos(phi + pi/4) + r**3*sin(phi)**2*sin(theta)**2*cos(theta) - r**3*sin(phi)**2*sin(theta)*cos(phi) + r**3*sin(phi)*sin(theta)**3 + r**2*(cos(phi - 2*theta) - cos(phi + 2*theta))/4 - r**2*sin(phi)**3*sin(theta)*cos(theta) + sqrt(2)*r**2*sin(phi)*sin(theta)**2*cos(phi + pi/4) - r**2*sin(2*phi)/2 + r**2*sin(theta)**2 + r*sin(phi)*sin(theta) + 1)*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)**2"
      ]
     },
     "execution_count": 208,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(t(om, u, v)) ; t(om, u, v).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>As for vectors and differential forms, the tensor components can be taken in any frame defined on the manifold:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{r^{3}}{\\left(r^{2}\\right)^{\\frac{3}{2}}} \\left(r^{3} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos^{2}{\\left (\\theta \\right )} + r^{2} \\sin^{2}{\\left (\\theta \\right )} \\cos^{2}{\\left (\\phi \\right )} + 1\\right) \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos^{2}{\\left (\\phi \\right )}</script></html>"
      ],
      "text/plain": [
       "r**3*(r**3*sin(phi)*sin(theta)*cos(theta)**2 + r**2*sin(theta)**2*cos(phi)**2 + 1)*sin(phi)*sin(theta)**3*cos(phi)**2/(r**2)**(3/2)"
      ]
     },
     "execution_count": 209,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t[Y.frame(), 1,1,1, Y]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Tensor calculus</h2>\n",
    "<p>The<strong> tensor product</strong> $\\otimes$ is denoted by `*`:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAUBAMAAADIGvgZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQ\nMqvFy5UvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJElEQVQoFXWSvUrEQBDHf0nuIyQXDQjXGt/A\njxeInd1tc/1Vcp2V+AT2VoJWVtaHcFiIkEblEDSPkMpKOQtRUeGcNRfIXsy/yMzO/NiZzCxYEbVa\n05kVuLm4rjBWfytlKZT4Nuxwqz1DD/hfdGKwFfYRvnimnuEELqEV4mS4P2YajmGaMoEuLGd4n4tA\nL9FA44x9eFR4H4uAnPcSfMUmrI4IvquAJzE7ZiiA+hdoZeAOOK0t0ZdLg1cNSJNupUnaKgekhLOO\nXflNDrFCvIFuUgbVjAU3FCjaIXbMWMJDJmnnzcjTvRo/QXPEucTvD15gNzGI3mz2Do1UjzqXWzhl\nTuYsy8rllxOFL8tiY364K4Ilq9etH8yforktG0eXtSL51Eie3C+wbD5oRkcPtwAAAABJRU5ErkJg\ngg==\n",
      "text/latex": [
       "$$\\left ( 0, \\quad 2\\right )$$"
      ],
      "text/plain": [
       "(0, 2)"
      ]
     },
     "execution_count": 210,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.tensor_type() ; a.tensor_type()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field v*A of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v\\otimes A</script></html>"
      ],
      "text/plain": [
       "Tensor field v*A of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 211,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = v*a ; print(b) ; b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The tensor product preserves the (anti)symmetries: since $A$ is a 2-form, it is antisymmetric with respect to its two arguments (positions 0 and 1); as a result, b is antisymmetric with respect to its last two arguments (positions 1 and 2):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "no symmetry; antisymmetry: (0, 1)\n"
     ]
    }
   ],
   "source": [
    "a.symmetries()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "no symmetry; antisymmetry: (1, 2)\n"
     ]
    }
   ],
   "source": [
    "b.symmetries()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Standard<strong> tensor arithmetics</strong> is implemented:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "s = - t + 2*f* b ; print(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p><strong>Tensor contractions</strong> are dealt with by the methods <span style=\"font-family: courier new,courier;\">trace()</span> and <span style=\"font-family: courier new,courier;\">contract()</span>: for instance, let us contract the tensor $T$ w.r.t. its first two arguments (positions 0 and 1), i.e. let us form the tensor $c$ of components $c_i = T^k_{\\ \\, k i}$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1-form on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "c = t.trace(0,1)\n",
    "print(c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>An alternative to the writing <span style=\"font-family: courier new,courier;\">trace(0,1)</span> is to use the <strong>index notation</strong> to denote the contraction: the indices are given in a string inside the [] operator, with '^' in front of the contravariant indices and '_' in front of the covariant ones:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1-form on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 216,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c1 = t['^k_ki']\n",
    "print(c1)\n",
    "c1 == c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The contraction is performed on the repeated index (here k); the letter denoting the remaining index (here i) is arbitrary:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 217,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t['^k_kj'] == c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 218,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t['^b_ba'] == c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>It can even be replaced by a dot:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 219,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t['^k_k.'] == c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>LaTeX notations are allowed:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 220,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t['^{k}_{ki}'] == c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The contraction $T^i_{\\ j k} v^k$ of the tensor fields $T$ and $v$ is taken as follows (2 refers to the last index position of $T$ and 0 to the only index position of v):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (1,1) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "tv = t.contract(2, v, 0)\n",
    "print(tv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Since 2 corresponds to the last index position of $T$ and 0 to the first index position of $v$, a shortcut for the above is</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (1,1) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "tv1 = t.contract(v)\n",
    "print(tv1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 223,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tv1 == tv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Instead of <span style=\"font-family: courier new,courier;\">contract()</span>, the <strong>index notation</strong>, combined with the <strong>*</strong> operator, can be used to denote the contraction:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 224,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t['^i_jk']*v['^k'] == tv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The non-repeated indices can be replaced by dots:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 225,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t['^._.k']*v['^k'] == tv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Metric structures</h2>\n",
    "<p>A <strong>Riemannian metric</strong> on the manifold $\\mathcal{M}$ is declared as follows:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Riemannian metric g on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "g = M.riemannian_metric('g')\n",
    "print(g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>It is a symmetric tensor field of type (0,2):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{T}^{(0, 2)}\\left(\\right)</script></html>"
      ],
      "text/plain": [
       "Free module T^(0,2)(M) of type-(0,2) tensors fields on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 227,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.parent()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Free module T^(0,2)(M) of type-(0,2) tensors fields on the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(g.parent())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "symmetry: (0, 1); no antisymmetry\n"
     ]
    }
   ],
   "source": [
    "g.symmetries()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The metric is initialized by its components with respect to some vector frame. For instance, using the default frame of $\\mathcal{M}$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\mathrm{d} x\\otimes \\mathrm{d} x+\\mathrm{d} y\\otimes \\mathrm{d} y+\\mathrm{d} z\\otimes \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "g = dx*dx + dy*dy + dz*dz"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g[1,1], g[2,2], g[3,3] = 1, 1, 1\n",
    "g.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The components w.r.t. another vector frame are obtained as for any tensor field:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin^{2}{\\left (\\theta \\right )} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "g = dr*dr + r**2 dtheta*dtheta + r**2*sin(theta)**2 dphi*dphi"
      ]
     },
     "execution_count": 231,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.display(Y.frame(), Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Of course, the metric acts on vector pairs:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field g(u,v) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g\\left(u,v\\right):& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x y z \\operatorname{u_{z}}{\\left (x,y,z \\right )} - x \\operatorname{u_{y}}{\\left (x,y,z \\right )} + y \\operatorname{u_{x}}{\\left (x,y,z \\right )} + \\operatorname{u_{x}}{\\left (x,y,z \\right )} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{3} \\operatorname{u_{z}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + r \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} - r \\operatorname{u_{y}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\operatorname{u_{x}}{\\left (r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )},r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )},r \\cos{\\left (\\theta \\right )} \\right )} \\end{array}</script></html>"
      ],
      "text/plain": [
       "g(u,v): U --> R\n",
       "   (x, y, z) |--> x*y*z*u_z(x, y, z) - x*u_y(x, y, z) + y*u_x(x, y, z) + u_x(x, y, z)\n",
       "   (r, theta, phi) |--> r**3*u_z(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta)**2*cos(phi)*cos(theta) + r*u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(phi)*sin(theta) - r*u_y(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))*sin(theta)*cos(phi) + u_x(r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta))"
      ]
     },
     "execution_count": 232,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.display() ; v.display(); print(g(u,v)) ; g(u,v).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The <strong>Levi-Civita connection</strong> associated to the metric $g$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Levi-Civita connection nabla_g associated with the Riemannian metric g on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\nabla_{g}</script></html>"
      ],
      "text/plain": [
       "Levi-Civita connection nabla_g associated with the Riemannian metric g on the 3-dimensional differentiable manifold M"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla = g.connection() \n",
    "print(nabla) ; nabla"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The Christoffel symbols with respect to the manifold's default coordinates:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[0, 0, 0\\right], \\left[0, 0, 0\\right], \\left[0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0\\right], \\left[0, 0, 0\\right], \\left[0, 0, 0\\right]\\right], \\left[\\left[0, 0, 0\\right], \\left[0, 0, 0\\right], \\left[0, 0, 0\\right]\\right]\\right]</script></html>"
      ],
      "text/plain": [
       "[[[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       " [[0, 0, 0], [0, 0, 0], [0, 0, 0]],\n",
       " [[0, 0, 0], [0, 0, 0], [0, 0, 0]]]"
      ]
     },
     "execution_count": 234,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla.coef()[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The Christoffel symbols with respect to the coordinates $(r,\\theta,\\phi)$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[0, 0, 0\\right], \\left[0, - r, 0\\right], \\left[0, 0, - r \\sin^{2}{\\left (\\theta \\right )}\\right]\\right], \\left[\\left[0, \\frac{1}{r}, 0\\right], \\left[\\frac{1}{r}, 0, 0\\right], \\left[0, 0, - \\frac{1}{2} \\sin{\\left (2 \\theta \\right )}\\right]\\right], \\left[\\left[0, 0, \\frac{1}{r}\\right], \\left[0, 0, \\frac{1}{\\tan{\\left (\\theta \\right )}}\\right], \\left[\\frac{1}{r}, \\frac{1}{\\tan{\\left (\\theta \\right )}}, 0\\right]\\right]\\right]</script></html>"
      ],
      "text/plain": [
       "[[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(theta)**2]],\n",
       " [[0, 1/r, 0], [1/r, 0, 0], [0, 0, -sin(2*theta)/2]],\n",
       " [[0, 0, 1/r], [0, 0, 1/tan(theta)], [1/r, 1/tan(theta), 0]]]"
      ]
     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla.coef(Y.frame())[:, Y]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "A nice view is obtained via the method `display()` (by default, only the nonzero connection coefficients are shown):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & - r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & - r \\sin^{2}{\\left (\\theta \\right )} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & - \\frac{1}{2} \\sin{\\left (2 \\theta \\right )} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{1}{\\tan{\\left (\\theta \\right )}} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & \\frac{1}{\\tan{\\left (\\theta \\right )}} \\end{array}</script></html>"
      ],
      "text/plain": [
       "Gam^r_theta,theta = -r \n",
       "Gam^r_phi,phi = -r*sin(theta)**2 \n",
       "Gam^theta_r,theta = 1/r \n",
       "Gam^theta_theta,r = 1/r \n",
       "Gam^theta_phi,phi = -sin(2*theta)/2 \n",
       "Gam^phi_r,phi = 1/r \n",
       "Gam^phi_theta,phi = 1/tan(theta) \n",
       "Gam^phi_phi,r = 1/r \n",
       "Gam^phi_phi,theta = 1/tan(theta) "
      ]
     },
     "execution_count": 236,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla.display(frame=Y.frame(), chart=Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The connection acting as a covariant derivative:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field nabla_g(v) of type (1,1) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-237-9cd3eeb069c4>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[0mnab_v\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnabla\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mv\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[1;32mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnab_v\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m;\u001b[0m \u001b[0mnab_v\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdisplay\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sage/tensor/modules/free_module_tensor.pyc\u001b[0m in \u001b[0;36mdisplay\u001b[1;34m(self, basis, format_spec)\u001b[0m\n\u001b[0;32m    665\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mind\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mcomp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mindex_generator\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    666\u001b[0m             \u001b[0mind_arg\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mind\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mformat_spec\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 667\u001b[1;33m             \u001b[0mcoef\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mcomp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mind_arg\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    668\u001b[0m             \u001b[1;31m# Check whether the coefficient is zero, preferably via\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    669\u001b[0m             \u001b[1;31m# the fast method is_trivial_zero():\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sage/tensor/modules/comp.pyc\u001b[0m in \u001b[0;36m__getitem__\u001b[1;34m(self, args)\u001b[0m\n\u001b[0;32m    740\u001b[0m                     \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_comp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mind\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    741\u001b[0m                 \u001b[1;32melif\u001b[0m \u001b[0mformat_type\u001b[0m \u001b[1;32mis\u001b[0m \u001b[0mNone\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 742\u001b[1;33m                     \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_output_formatter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_comp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mind\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    743\u001b[0m                 \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    744\u001b[0m                     \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_output_formatter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_comp\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mind\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mformat_type\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sage/manifolds/scalarfield.pyc\u001b[0m in \u001b[0;36mcoord_function\u001b[1;34m(self, chart, from_chart)\u001b[0m\n\u001b[0;32m   1571\u001b[0m             coords = [ change._transf._functions[i].expr()\n\u001b[0;32m   1572\u001b[0m                        for i in range(self._manifold.dim()) ]\n\u001b[1;32m-> 1573\u001b[1;33m             \u001b[0mnew_expr\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_express\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mfrom_chart\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mcoords\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1574\u001b[0m             \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_express\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mchart\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mchart\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mfunction\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mnew_expr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1575\u001b[0m             \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_del_derived\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sage/manifolds/chart_func.pyc\u001b[0m in \u001b[0;36m__call__\u001b[1;34m(self, *coords, **options)\u001b[0m\n\u001b[0;32m    697\u001b[0m                 \u001b[1;32mreturn\u001b[0m \u001b[0mresu\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    698\u001b[0m         \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 699\u001b[1;33m             \u001b[1;32mreturn\u001b[0m \u001b[0mcalc\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msimplify\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mresu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mcurr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    700\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    701\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0m__bool__\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sage/manifolds/calculus_method.pyc\u001b[0m in \u001b[0;36msimplify\u001b[1;34m(self, expression, method)\u001b[0m\n\u001b[0;32m    234\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mmethod\u001b[0m \u001b[1;32mis\u001b[0m \u001b[0mNone\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    235\u001b[0m             \u001b[0mmethod\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_current\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 236\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_simplify_dict\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mmethod\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpression\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    237\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    238\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0mis_trivial_zero\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mexpression\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mNone\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sage/manifolds/utilities.pyc\u001b[0m in \u001b[0;36msimplify_chain_real_sympy\u001b[1;34m(expr)\u001b[0m\n\u001b[0;32m    545\u001b[0m     \"\"\"\n\u001b[0;32m    546\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcombsimp\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 547\u001b[1;33m     \u001b[0mexpr\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtrigsimp\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    548\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[1;34m'sqrt('\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mstr\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    549\u001b[0m         \u001b[0mexpr\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0msimplify_sqrt_real\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_sympy_\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/core/expr.pyc\u001b[0m in \u001b[0;36mtrigsimp\u001b[1;34m(self, **args)\u001b[0m\n\u001b[0;32m   3046\u001b[0m         \u001b[1;34m\"\"\"See the trigsimp function in sympy.simplify\"\"\"\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   3047\u001b[0m         \u001b[1;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msimplify\u001b[0m \u001b[1;32mimport\u001b[0m \u001b[0mtrigsimp\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 3048\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mtrigsimp\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   3049\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   3050\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0mradsimp\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/trigsimp.pyc\u001b[0m in \u001b[0;36mtrigsimp\u001b[1;34m(expr, **opts)\u001b[0m\n\u001b[0;32m    511\u001b[0m                    }[method]\n\u001b[0;32m    512\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 513\u001b[1;33m     \u001b[1;32mreturn\u001b[0m \u001b[0mtrigsimpfunc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    514\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    515\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/trigsimp.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[1;34m(x)\u001b[0m\n\u001b[0;32m    504\u001b[0m     trigsimpfunc = {\n\u001b[0;32m    505\u001b[0m         \u001b[1;34m'fu'\u001b[0m\u001b[1;33m:\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m:\u001b[0m \u001b[0mfu\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mopts\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 506\u001b[1;33m         \u001b[1;34m'matching'\u001b[0m\u001b[1;33m:\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m:\u001b[0m \u001b[0mfutrig\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    507\u001b[0m         \u001b[1;34m'groebner'\u001b[0m\u001b[1;33m:\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m:\u001b[0m \u001b[0mgroebnersimp\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mopts\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    508\u001b[0m         'combined': (lambda x: futrig(groebnersimp(x,\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/trigsimp.pyc\u001b[0m in \u001b[0;36mfutrig\u001b[1;34m(e, **kwargs)\u001b[0m\n\u001b[0;32m   1081\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1082\u001b[0m     \u001b[0mold\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0me\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1083\u001b[1;33m     \u001b[0me\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mbottom_up\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0me\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m:\u001b[0m \u001b[0m_futrig\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1084\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1085\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpop\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'hyper'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mTrue\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mand\u001b[0m \u001b[0me\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mhas\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mHyperbolicFunction\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/simplify.pyc\u001b[0m in \u001b[0;36mbottom_up\u001b[1;34m(rv, F, atoms, nonbasic)\u001b[0m\n\u001b[0;32m    925\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mrv\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    926\u001b[0m             args = tuple([bottom_up(a, F, atoms, nonbasic)\n\u001b[1;32m--> 927\u001b[1;33m                 for a in rv.args])\n\u001b[0m\u001b[0;32m    928\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0margs\u001b[0m \u001b[1;33m!=\u001b[0m \u001b[0mrv\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    929\u001b[0m                 \u001b[0mrv\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mrv\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/simplify.pyc\u001b[0m in \u001b[0;36mbottom_up\u001b[1;34m(rv, F, atoms, nonbasic)\u001b[0m\n\u001b[0;32m    928\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0margs\u001b[0m \u001b[1;33m!=\u001b[0m \u001b[0mrv\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    929\u001b[0m                 \u001b[0mrv\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mrv\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 930\u001b[1;33m             \u001b[0mrv\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mF\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mrv\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    931\u001b[0m         \u001b[1;32melif\u001b[0m \u001b[0matoms\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    932\u001b[0m             \u001b[0mrv\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mF\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mrv\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/trigsimp.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[1;34m(x)\u001b[0m\n\u001b[0;32m   1081\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1082\u001b[0m     \u001b[0mold\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0me\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1083\u001b[1;33m     \u001b[0me\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mbottom_up\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0me\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m:\u001b[0m \u001b[0m_futrig\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1084\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1085\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpop\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'hyper'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mTrue\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mand\u001b[0m \u001b[0me\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mhas\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mHyperbolicFunction\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/trigsimp.pyc\u001b[0m in \u001b[0;36m_futrig\u001b[1;34m(e, **kwargs)\u001b[0m\n\u001b[0;32m   1149\u001b[0m             factor_terms, TR12(x), trigs)],  # expand tan of sum\n\u001b[0;32m   1150\u001b[0m         )]\n\u001b[1;32m-> 1151\u001b[1;33m     \u001b[0me\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mgreedy\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mtree\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mobjective\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mLops\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0me\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1152\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1153\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mcoeff\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0me\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/strategies/core.pyc\u001b[0m in \u001b[0;36mminrule\u001b[1;34m(expr)\u001b[0m\n\u001b[0;32m    115\u001b[0m     \u001b[0mobjective\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mkwargs\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'objective'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0midentity\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    116\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0mminrule\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 117\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mrule\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mrule\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrules\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mkey\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mobjective\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    118\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mminrule\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/strategies/core.pyc\u001b[0m in \u001b[0;36mchain_rl\u001b[1;34m(expr)\u001b[0m\n\u001b[0;32m     42\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0mchain_rl\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     43\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mrule\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrules\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 44\u001b[1;33m             \u001b[0mexpr\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mrule\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     45\u001b[0m         \u001b[1;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     46\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mchain_rl\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/trigsimp.pyc\u001b[0m in \u001b[0;36m<lambda>\u001b[1;34m(x)\u001b[0m\n\u001b[0;32m   1126\u001b[0m         \u001b[0mTR10\u001b[0m\u001b[1;33m,\u001b[0m  \u001b[1;31m# sin-cos of sums -> sin-cos prod\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1127\u001b[0m         \u001b[0mTR11\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mTR6\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;31m# reduce double angles and rewrite cos pows\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1128\u001b[1;33m         \u001b[1;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m:\u001b[0m \u001b[0m_eapply\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mfactor\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mtrigs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1129\u001b[0m         \u001b[0mTR14\u001b[0m\u001b[1;33m,\u001b[0m  \u001b[1;31m# factored powers of identities\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1130\u001b[0m         \u001b[1;33m[\u001b[0m\u001b[0midentity\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m:\u001b[0m \u001b[0m_eapply\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0m_mexpand\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mtrigs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/simplify/trigsimp.pyc\u001b[0m in \u001b[0;36m_eapply\u001b[1;34m(func, e, cond)\u001b[0m\n\u001b[0;32m   1168\u001b[0m         \u001b[1;32mreturn\u001b[0m \u001b[0me\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1169\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0m_is_Expr\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0me\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mor\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[0me\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1170\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mfunc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0me\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1171\u001b[0m     return e.func(*[\n\u001b[0;32m   1172\u001b[0m         \u001b[0m_eapply\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mfunc\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mei\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mif\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mcond\u001b[0m \u001b[1;32mis\u001b[0m \u001b[0mNone\u001b[0m \u001b[1;32mor\u001b[0m \u001b[0mcond\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mei\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32melse\u001b[0m \u001b[0mei\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/polytools.pyc\u001b[0m in \u001b[0;36mfactor\u001b[1;34m(f, *gens, **args)\u001b[0m\n\u001b[0;32m   6061\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   6062\u001b[0m     \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 6063\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0m_generic_factor\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mgens\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;34m'factor'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   6064\u001b[0m     \u001b[1;32mexcept\u001b[0m \u001b[0mPolynomialError\u001b[0m \u001b[1;32mas\u001b[0m \u001b[0mmsg\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   6065\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mis_commutative\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/polytools.pyc\u001b[0m in \u001b[0;36m_generic_factor\u001b[1;34m(expr, gens, args, method)\u001b[0m\n\u001b[0;32m   5753\u001b[0m     \u001b[0moptions\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mallowed_flags\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   5754\u001b[0m     \u001b[0mopt\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mbuild_options\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mgens\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 5755\u001b[1;33m     \u001b[1;32mreturn\u001b[0m \u001b[0m_symbolic_factor\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0msympify\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   5756\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   5757\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/polytools.pyc\u001b[0m in \u001b[0;36m_symbolic_factor\u001b[1;34m(expr, opt, method)\u001b[0m\n\u001b[0;32m   5698\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;34m'_eval_factor'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   5699\u001b[0m             \u001b[1;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_eval_factor\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 5700\u001b[1;33m         \u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mfactors\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0m_symbolic_factor_list\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mtogether\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   5701\u001b[0m         \u001b[1;32mreturn\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0m_factors_product\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mfactors\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   5702\u001b[0m     \u001b[1;32melif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m'args'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/polytools.pyc\u001b[0m in \u001b[0;36m_symbolic_factor_list\u001b[1;34m(expr, opt, method)\u001b[0m\n\u001b[0;32m   5666\u001b[0m             \u001b[0mfunc\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mpoly\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmethod\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;34m'_list'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   5667\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 5668\u001b[1;33m             \u001b[0m_coeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0m_factors\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   5669\u001b[0m             \u001b[1;32mif\u001b[0m \u001b[0m_coeff\u001b[0m \u001b[1;32mis\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[0mS\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   5670\u001b[0m                 \u001b[1;32mif\u001b[0m \u001b[0mexp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mis_Integer\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/polytools.pyc\u001b[0m in \u001b[0;36mfactor_list\u001b[1;34m(f)\u001b[0m\n\u001b[0;32m   3097\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrep\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;34m'factor_list'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   3098\u001b[0m             \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 3099\u001b[1;33m                 \u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mfactors\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrep\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mfactor_list\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   3100\u001b[0m             \u001b[1;32mexcept\u001b[0m \u001b[0mDomainError\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   3101\u001b[0m                 \u001b[1;32mreturn\u001b[0m \u001b[0mS\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/polyclasses.pyc\u001b[0m in \u001b[0;36mfactor_list\u001b[1;34m(f)\u001b[0m\n\u001b[0;32m    757\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0mfactor_list\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    758\u001b[0m         \u001b[1;34m\"\"\"Returns a list of irreducible factors of ``f``. \"\"\"\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 759\u001b[1;33m         \u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mfactors\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_factor_list\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrep\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlev\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdom\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    760\u001b[0m         \u001b[1;32mreturn\u001b[0m \u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m[\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mper\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mk\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32mfor\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mk\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mfactors\u001b[0m \u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    761\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/factortools.pyc\u001b[0m in \u001b[0;36mdmp_factor_list\u001b[1;34m(f, u, K0)\u001b[0m\n\u001b[0;32m   1277\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mis_ZZ\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1278\u001b[0m             \u001b[0mlevels\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_exclude\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1279\u001b[1;33m             \u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mfactors\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_zz_factor\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1280\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1281\u001b[0m             \u001b[1;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mk\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mfactors\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/factortools.pyc\u001b[0m in \u001b[0;36mdmp_zz_factor\u001b[1;34m(f, u, K)\u001b[0m\n\u001b[0;32m   1089\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mdmp_degree\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1090\u001b[0m         \u001b[0mg\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_sqf_part\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1091\u001b[1;33m         \u001b[0mH\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_zz_wang\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1092\u001b[0m         \u001b[0mfactors\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_trial_division\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mH\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1093\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/factortools.pyc\u001b[0m in \u001b[0;36mdmp_zz_wang\u001b[1;34m(f, u, K, mod, seed)\u001b[0m\n\u001b[0;32m   1012\u001b[0m     \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1013\u001b[0m         \u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mH\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mLC\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_zz_wang_lead_coeffs\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mT\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mE\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mH\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mA\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1014\u001b[1;33m         \u001b[0mfactors\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_zz_wang_hensel_lifting\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mH\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mLC\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mA\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mp\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1015\u001b[0m     \u001b[1;32mexcept\u001b[0m \u001b[0mExtraneousFactors\u001b[0m\u001b[1;33m:\u001b[0m  \u001b[1;31m# pragma: no cover\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1016\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mquery\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'EEZ_RESTART_IF_NEEDED'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/factortools.pyc\u001b[0m in \u001b[0;36mdmp_zz_wang_hensel_lifting\u001b[1;34m(f, H, LC, A, p, u, K)\u001b[0m\n\u001b[0;32m    882\u001b[0m                     \u001b[0mH\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_ground_trunc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mh\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mp\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mw\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    883\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 884\u001b[1;33m                 \u001b[0mh\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_sub\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0ms\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_expand\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mH\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mw\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mw\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    885\u001b[0m                 \u001b[0mc\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_ground_trunc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mh\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mp\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mw\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    886\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_expand\u001b[1;34m(polys, u, K)\u001b[0m\n\u001b[0;32m   1829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1830\u001b[0m     \u001b[1;32mfor\u001b[0m \u001b[0mg\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mpolys\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1831\u001b[1;33m         \u001b[0mf\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1833\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_mul\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    830\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmax\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 831\u001b[1;33m             \u001b[0mcoeff\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    833\u001b[0m         \u001b[0mh\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_mul\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    830\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmax\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 831\u001b[1;33m             \u001b[0mcoeff\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    833\u001b[0m         \u001b[0mh\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_mul\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    830\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmax\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 831\u001b[1;33m             \u001b[0mcoeff\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    833\u001b[0m         \u001b[0mh\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_mul\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    830\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmax\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 831\u001b[1;33m             \u001b[0mcoeff\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    833\u001b[0m         \u001b[0mh\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_mul\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    830\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmax\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 831\u001b[1;33m             \u001b[0mcoeff\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    833\u001b[0m         \u001b[0mh\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_mul\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    830\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmax\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 831\u001b[1;33m             \u001b[0mcoeff\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    833\u001b[0m         \u001b[0mh\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_mul\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    829\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    830\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mmax\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mi\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 831\u001b[1;33m             \u001b[0mcoeff\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdmp_mul\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mi\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mj\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mv\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    832\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    833\u001b[0m         \u001b[0mh\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdmp_add\u001b[1;34m(f, g, u, K)\u001b[0m\n\u001b[0;32m    564\u001b[0m     \"\"\"\n\u001b[0;32m    565\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 566\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mdup_add\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mK\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    567\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    568\u001b[0m     \u001b[0mdf\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdmp_degree\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densearith.pyc\u001b[0m in \u001b[0;36mdup_add\u001b[1;34m(f, g, K)\u001b[0m\n\u001b[0;32m    534\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    535\u001b[0m     \u001b[0mdf\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdup_degree\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 536\u001b[1;33m     \u001b[0mdg\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mdup_degree\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    537\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    538\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mdf\u001b[0m \u001b[1;33m==\u001b[0m \u001b[0mdg\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/eric/sage/8.1.beta8/local/lib/python2.7/site-packages/sympy/polys/densebasic.pyc\u001b[0m in \u001b[0;36mdup_degree\u001b[1;34m(f)\u001b[0m\n\u001b[0;32m    155\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[1;32mnot\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    156\u001b[0m         \u001b[1;32mreturn\u001b[0m \u001b[1;33m-\u001b[0m\u001b[0moo\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 157\u001b[1;33m     \u001b[1;32mreturn\u001b[0m \u001b[0mlen\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;33m-\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    158\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    159\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32msrc/cysignals/signals.pyx\u001b[0m in \u001b[0;36mcysignals.signals.python_check_interrupt\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;32msrc/cysignals/signals.pyx\u001b[0m in \u001b[0;36mcysignals.signals.sig_raise_exception\u001b[1;34m()\u001b[0m\n",
      "\u001b[1;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "nab_v = nabla(v)\n",
    "print(nab_v)\n",
    "# nab_v.display()  # does not work with SymPy (trigsimp runs forever...)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Being a Levi-Civita connection, $\\nabla_g$ is torsion.free:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (1,2) on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 238,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(nabla.torsion()) ; nabla.torsion().display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In the present case, it is also flat:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field Riem(g) of type (1,3) on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = 0</script></html>"
      ],
      "text/plain": [
       "Riem(g) = 0"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(nabla.riemann()) ; nabla.riemann().display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Let us consider a non-flat metric, by changing $g_{rr}$ to $1/(1+r^2)$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin^{2}{\\left (\\theta \\right )} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "g = 1/(r**2 + 1) dr*dr + r**2 dtheta*dtheta + r**2*sin(theta)**2 dphi*dphi"
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g[Y.frame(), 1,1, Y] = 1/(1+r^2)\n",
    "g.display(Y.frame(), Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>For convenience, we change the default chart on the domain $U$ to Y=$(U,(r,\\theta,\\phi))$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "U.set_default_chart(Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In this way, we do not have to specify Y when asking for coordinate expressions in terms of $(r,\\theta,\\phi)$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + r^{2} \\sin^{2}{\\left (\\theta \\right )} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "g = 1/(r**2 + 1) dr*dr + r**2 dtheta*dtheta + r**2*sin(theta)**2 dphi*dphi"
      ]
     },
     "execution_count": 242,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.display(Y.frame())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>We recognize the metric of the hyperbolic space $\\mathbb{H}^3$. Its expression in terms of the chart $(U,(x,y,z))$ is</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{y^{2} + z^{2} + 1}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} x + \\left( - \\frac{x y}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( - \\frac{x z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} z + \\left( - \\frac{x y}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} x + \\left( \\frac{x^{2} + z^{2} + 1}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( - \\frac{y z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} z + \\left( - \\frac{x z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} z\\otimes \\mathrm{d} x + \\left( - \\frac{y z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} z\\otimes \\mathrm{d} y + \\left( \\frac{x^{2} + y^{2} + 1}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} z\\otimes \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "g = (y**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) dx*dx - x*y/(x**2 + y**2 + z**2 + 1) dx*dy - x*z/(x**2 + y**2 + z**2 + 1) dx*dz - x*y/(x**2 + y**2 + z**2 + 1) dy*dx + (x**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) dy*dy - y*z/(x**2 + y**2 + z**2 + 1) dy*dz - x*z/(x**2 + y**2 + z**2 + 1) dz*dx - y*z/(x**2 + y**2 + z**2 + 1) dz*dy + (x**2 + y**2 + 1)/(x**2 + y**2 + z**2 + 1) dz*dz"
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.display(X_U.frame(), X_U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>A matrix view of the components may be more appropriate:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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tmNSiKkMCeNqoadQe5X7SgmEQZo2exyNeEJSoC2KiRS/7XeEkDHwXpNq9Q/RO7trvimZ5\n9MjRXq9spQUtGG1fXPQmwpiUXtxLDqUTUUbQa4emA8Rwb68cSofW/a510zUCbygYd5/PSBPaGtAV\nfhakAd1vBWVjKjLTK6jd4NrtIhDgkANFsxTEVI1cu11klofgL4EY04LOBIlJwBPEJAS8rKKrzjSB\ngHXTNQoPnR448xkRqqyONtQe5f7wu6ALrHxMxQAT8u01QMCoWR5jWnD1tL0T6RqVHrmiuDurpG2U\n+9iCN2zfA3Dgjm3xb9DLx1RwcRfJ1gABI2Z5zLbvPs76bzhj7/TCFca0QtPe/3XTNQLP+1q4WCVt\nI9y31XXf9ik4MX+ja8H5/GRhIO8itY35nG9Bw9m7iIiyUCe6ynhyWpauW+ev/x/3demg6vjgJgAA\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left [ \\frac{y^{2} + z^{2} + 1}{x^{2} + y^{2} + z^{2} + 1}, \\quad - \\frac{x y}{x^{2} + y^{2} + z^{2} + 1}, \\quad - \\frac{x z}{x^{2} + y^{2} + z^{2} + 1}\\right ], \\quad \\left [ - \\frac{x y}{x^{2} + y^{2} + z^{2} + 1}, \\quad \\frac{x^{2} + z^{2} + 1}{x^{2} + y^{2} + z^{2} + 1}, \\quad - \\frac{y z}{x^{2} + y^{2} + z^{2} + 1}\\right ], \\quad \\left [ - \\frac{x z}{x^{2} + y^{2} + z^{2} + 1}, \\quad - \\frac{y z}{x^{2} + y^{2} + z^{2} + 1}, \\quad \\frac{x^{2} + y^{2} + 1}{x^{2} + y^{2} + z^{2} + 1}\\right ]\\right ]$$"
      ],
      "text/plain": [
       "[[(y**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1),\n",
       "  -x*y/(x**2 + y**2 + z**2 + 1),\n",
       "  -x*z/(x**2 + y**2 + z**2 + 1)],\n",
       " [-x*y/(x**2 + y**2 + z**2 + 1),\n",
       "  (x**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1),\n",
       "  -y*z/(x**2 + y**2 + z**2 + 1)],\n",
       " [-x*z/(x**2 + y**2 + z**2 + 1),\n",
       "  -y*z/(x**2 + y**2 + z**2 + 1),\n",
       "  (x**2 + y**2 + 1)/(x**2 + y**2 + z**2 + 1)]]"
      ]
     },
     "execution_count": 244,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g[X_U.frame(), :, X_U]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>We extend these components, a priori defined only on $U$, to the whole manifold $\\mathcal{M}$, by demanding the same coordinate expressions in the frame associated to the chart X=$(\\mathcal{M},(x,y,z))$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{y^{2} + z^{2} + 1}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} x + \\left( - \\frac{x y}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( - \\frac{x z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} x\\otimes \\mathrm{d} z + \\left( - \\frac{x y}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} x + \\left( \\frac{x^{2} + z^{2} + 1}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} y + \\left( - \\frac{y z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} y\\otimes \\mathrm{d} z + \\left( - \\frac{x z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} z\\otimes \\mathrm{d} x + \\left( - \\frac{y z}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} z\\otimes \\mathrm{d} y + \\left( \\frac{x^{2} + y^{2} + 1}{x^{2} + y^{2} + z^{2} + 1} \\right) \\mathrm{d} z\\otimes \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "g = (y**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) dx*dx - x*y/(x**2 + y**2 + z**2 + 1) dx*dy - x*z/(x**2 + y**2 + z**2 + 1) dx*dz - x*y/(x**2 + y**2 + z**2 + 1) dy*dx + (x**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) dy*dy - y*z/(x**2 + y**2 + z**2 + 1) dy*dz - x*z/(x**2 + y**2 + z**2 + 1) dz*dx - y*z/(x**2 + y**2 + z**2 + 1) dz*dy + (x**2 + y**2 + 1)/(x**2 + y**2 + z**2 + 1) dz*dz"
      ]
     },
     "execution_count": 245,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.add_comp_by_continuation(X.frame(), U, X)\n",
    "g.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The Levi-Civita connection is automatically recomputed, after the change in $g$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "nabla = g.connection()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In particular, the Christoffel symbols are different:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, x} \\, x \\, x }^{ \\, x \\phantom{\\, x} \\phantom{\\, x} } & = & - \\frac{x \\left(y^{2} + z^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, x \\, y }^{ \\, x \\phantom{\\, x} \\phantom{\\, y} } & = & \\frac{x^{2} y}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, x \\, z }^{ \\, x \\phantom{\\, x} \\phantom{\\, z} } & = & \\frac{x^{2} z}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, y \\, y }^{ \\, x \\phantom{\\, y} \\phantom{\\, y} } & = & - \\frac{x \\left(x^{2} + z^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, y \\, z }^{ \\, x \\phantom{\\, y} \\phantom{\\, z} } & = & \\frac{x y z}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, x} \\, z \\, z }^{ \\, x \\phantom{\\, z} \\phantom{\\, z} } & = & - \\frac{x \\left(x^{2} + y^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, x }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} } & = & - \\frac{y \\left(y^{2} + z^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, y }^{ \\, y \\phantom{\\, x} \\phantom{\\, y} } & = & \\frac{x y^{2}}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, z }^{ \\, y \\phantom{\\, x} \\phantom{\\, z} } & = & \\frac{x y z}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, y }^{ \\, y \\phantom{\\, y} \\phantom{\\, y} } & = & - \\frac{y \\left(x^{2} + z^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, z }^{ \\, y \\phantom{\\, y} \\phantom{\\, z} } & = & \\frac{y^{2} z}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, y} \\, z \\, z }^{ \\, y \\phantom{\\, z} \\phantom{\\, z} } & = & - \\frac{y \\left(x^{2} + y^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, z} \\, x \\, x }^{ \\, z \\phantom{\\, x} \\phantom{\\, x} } & = & - \\frac{z \\left(y^{2} + z^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, z} \\, x \\, y }^{ \\, z \\phantom{\\, x} \\phantom{\\, y} } & = & \\frac{x y z}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, z} \\, x \\, z }^{ \\, z \\phantom{\\, x} \\phantom{\\, z} } & = & \\frac{x z^{2}}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, z} \\, y \\, y }^{ \\, z \\phantom{\\, y} \\phantom{\\, y} } & = & - \\frac{z \\left(x^{2} + z^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, z} \\, y \\, z }^{ \\, z \\phantom{\\, y} \\phantom{\\, z} } & = & \\frac{y z^{2}}{x^{2} + y^{2} + z^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, z} \\, z \\, z }^{ \\, z \\phantom{\\, z} \\phantom{\\, z} } & = & - \\frac{z \\left(x^{2} + y^{2} + 1\\right)}{x^{2} + y^{2} + z^{2} + 1} \\end{array}</script></html>"
      ],
      "text/plain": [
       "Gam^x_xx = -x*(y**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^x_xy = x**2*y/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^x_xz = x**2*z/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^x_yy = -x*(x**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^x_yz = x*y*z/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^x_zz = -x*(x**2 + y**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^y_xx = -y*(y**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^y_xy = x*y**2/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^y_xz = x*y*z/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^y_yy = -y*(x**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^y_yz = y**2*z/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^y_zz = -y*(x**2 + y**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^z_xx = -z*(y**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^z_xy = x*y*z/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^z_xz = x*z**2/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^z_yy = -z*(x**2 + z**2 + 1)/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^z_yz = y*z**2/(x**2 + y**2 + z**2 + 1) \n",
       "Gam^z_zz = -z*(x**2 + y**2 + 1)/(x**2 + y**2 + z**2 + 1) "
      ]
     },
     "execution_count": 247,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla.display(only_nonredundant=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & - \\frac{r}{r^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & - r \\left(r^{2} + 1\\right) \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & - r \\left(r^{2} + 1\\right) \\sin^{2}{\\left (\\theta \\right )} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & - \\frac{1}{2} \\sin{\\left (2 \\theta \\right )} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{1}{\\tan{\\left (\\theta \\right )}} \\end{array}</script></html>"
      ],
      "text/plain": [
       "Gam^r_r,r = -r/(r**2 + 1) \n",
       "Gam^r_theta,theta = -r*(r**2 + 1) \n",
       "Gam^r_phi,phi = -r*(r**2 + 1)*sin(theta)**2 \n",
       "Gam^theta_r,theta = 1/r \n",
       "Gam^theta_phi,phi = -sin(2*theta)/2 \n",
       "Gam^phi_r,phi = 1/r \n",
       "Gam^phi_theta,phi = 1/tan(theta) "
      ]
     },
     "execution_count": 248,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nabla.display(frame=Y.frame(), chart=Y, only_nonredundant=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The <strong>Riemann tensor</strong> is now</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field Riem(g) of type (1,3) on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = - r^{2} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + r^{2} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r - r^{2} \\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + r^{2} \\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + \\left( \\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta} + \\left( - \\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} r - r^{2} \\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} + r^{2} \\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial {\\theta} }\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta} + \\left( \\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi} + \\left( - \\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} r + r^{2} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi} - r^{2} \\frac{\\partial}{\\partial {\\phi} }\\otimes \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\theta}</script></html>"
      ],
      "text/plain": [
       "Riem(g) = -r**2 d/dr*dtheta*dr*dtheta + r**2 d/dr*dtheta*dtheta*dr - r**2*sin(theta)**2 d/dr*dphi*dr*dphi + r**2*sin(theta)**2 d/dr*dphi*dphi*dr + 1/(r**2 + 1) d/dtheta*dr*dr*dtheta - 1/(r**2 + 1) d/dtheta*dr*dtheta*dr - r**2*sin(theta)**2 d/dtheta*dphi*dtheta*dphi + r**2*sin(theta)**2 d/dtheta*dphi*dphi*dtheta + 1/(r**2 + 1) d/dphi*dr*dr*dphi - 1/(r**2 + 1) d/dphi*dr*dphi*dr + r**2 d/dphi*dtheta*dtheta*dphi - r**2 d/dphi*dtheta*dphi*dtheta"
      ]
     },
     "execution_count": 249,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Riem = nabla.riemann()\n",
    "print(Riem) ; Riem.display(Y.frame())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Note that it can be accessed directely via the metric, without any explicit mention of the connection:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 250,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.riemann() is nabla.riemann()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The <strong>Ricci tensor</strong> is</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Field of symmetric bilinear forms Ric(g) on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = \\left( - \\frac{2}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r - 2 r^{2} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} - 2 r^{2} \\sin^{2}{\\left (\\theta \\right )} \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "Ric(g) = -2/(r**2 + 1) dr*dr - 2*r**2 dtheta*dtheta - 2*r**2*sin(theta)**2 dphi*dphi"
      ]
     },
     "execution_count": 251,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ric = g.ricci()\n",
    "print(Ric) ; Ric.display(Y.frame())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The<strong> Weyl tensor</strong> is:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field C(g) of type (1,3) on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{C}\\left(g\\right) = 0</script></html>"
      ],
      "text/plain": [
       "C(g) = 0"
      ]
     },
     "execution_count": 252,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = g.weyl()\n",
    "print(C) ; C.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The Weyl tensor vanishes identically because the dimension of $\\mathcal{M}$ is 3.</p>\n",
    "<p>Finally, the <strong>Ricci scalar</strong> is </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field r(g) on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathcal{M} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & -6 \\\\ \\mbox{on}\\ U : & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & -6 \\end{array}</script></html>"
      ],
      "text/plain": [
       "r(g): M --> R\n",
       "   (x, y, z) |--> -6\n",
       "on U: (r, theta, phi) |--> -6"
      ]
     },
     "execution_count": 253,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = g.ricci_scalar()\n",
    "print(R) ; R.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>We recover the fact that $\\mathbb{H}^3$ is a Riemannian manifold of constant negative curvature.</p>\n",
    "\n",
    "<h2>Tensor transformations induced by a metric</h2>\n",
    "<p>The most important tensor transformation induced by the metric $g$ is the so-called <strong>musical isomorphism</strong>, or <strong>index raising</strong> and<strong> index lowering</strong>:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field T of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 255,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T = \\left( r^{2} \\sin^{2}{\\left (\\theta \\right )} \\cos^{2}{\\left (\\phi \\right )} + 1 \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x + r^{3} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} \\frac{\\partial}{\\partial z }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x</script></html>"
      ],
      "text/plain": [
       "T = (r**2*sin(theta)**2*cos(phi)**2 + 1) d/dx*dy*dx + r**3*sin(phi)*sin(theta)**2*cos(phi)*cos(theta) d/dz*dy*dx"
      ]
     },
     "execution_count": 255,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 256,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}T = \\left( x^{2} + 1 \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x + x y z \\frac{\\partial}{\\partial z }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x</script></html>"
      ],
      "text/plain": [
       "T = (x**2 + 1) d/dx*dy*dx + x*y*z d/dz*dy*dx"
      ]
     },
     "execution_count": 256,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.display(X_U.frame(), X_U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Raising the last index of $T$ with $g$:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 257,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (2,1) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "s = t.up(g, 2)\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Raising all the covariant indices of $T$ (i.e. those at the positions 1 and 2):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 258,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (3,0) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "s = t.up(g)\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 259,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (0,3) on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "s = t.down(g)\n",
    "print(s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Hodge duality</h2>\n",
    "<p>The volume 3-form (Levi-Civita tensor) associated with the metric $g$ is</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 260,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3-form eps_g on the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\epsilon_{g} = \\left( \\frac{1}{\\sqrt{x^{2} + y^{2} + z^{2} + 1}} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "eps_g = 1/sqrt(x**2 + y**2 + z**2 + 1) dx/\\dy/\\dz"
      ]
     },
     "execution_count": 260,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon = g.volume_form()\n",
    "print(epsilon) ; epsilon.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 261,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\epsilon_{g} = \\left( \\frac{r^{2} \\sin{\\left (\\theta \\right )}}{\\sqrt{r^{2} + 1}} \\right) \\mathrm{d} r\\wedge \\mathrm{d} {\\theta}\\wedge \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "eps_g = r**2*sin(theta)/sqrt(r**2 + 1) dr/\\dtheta/\\dphi"
      ]
     },
     "execution_count": 261,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon.display(Y.frame())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 262,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field f on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x + y^{2} + z^{3} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: U --> R\n",
       "   (x, y, z) |--> x + y**2 + z**3\n",
       "   (r, theta, phi) |--> r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))"
      ]
     },
     "execution_count": 262,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(f) ; f.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 263,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3-form *f on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\star f = \\left( \\frac{r}{\\sqrt{r^{2} + 1}} \\left(r^{2} \\cos^{3}{\\left (\\theta \\right )} + r \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\right) \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "*f = r*(r**2*cos(theta)**3 + r*sin(phi)**2*sin(theta)**2 + sin(theta)*cos(phi))/sqrt(r**2 + 1) dx/\\dy/\\dz"
      ]
     },
     "execution_count": 263,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sf = f.hodge_dual(g)\n",
    "print(sf) ; sf.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>We check the classical formula $\\star f = f\\, \\epsilon_g$, or, more precisely, $\\star f = f\\, \\epsilon_g|_U$ (for $f$ is defined on $U$ only):</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 264,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 264,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sf == f * epsilon.restrict(U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The Hodge dual of a 1-form is a 2-form:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 265,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1-form omega on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\omega = r^{2} \\sin^{2}{\\left (\\theta \\right )} \\mathrm{d} x + r \\cos{\\left (\\theta \\right )} \\mathrm{d} y + r \\left(\\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\cos{\\left (\\theta \\right )}\\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "omega = r**2*sin(theta)**2 dx + r*cos(theta) dy + r*(sin(theta)*cos(phi) - cos(theta)) dz"
      ]
     },
     "execution_count": 265,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(om) ; om.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 266,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2-form *omega on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\star \\omega = \\left( \\frac{r}{\\sqrt{r^{2} + 1}} \\left(r^{3} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} - \\sqrt{2} r^{2} \\sin^{3}{\\left (\\theta \\right )} \\sin{\\left (\\phi + \\frac{\\pi}{4} \\right )} + \\sqrt{2} r^{2} \\sin{\\left (\\theta \\right )} \\sin{\\left (\\phi + \\frac{\\pi}{4} \\right )} - r^{2} \\cos^{3}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\cos{\\left (\\theta \\right )}\\right) \\right) \\mathrm{d} x\\wedge \\mathrm{d} y + \\left( \\frac{r}{\\sqrt{r^{2} + 1}} \\left(- \\frac{r^{3}}{8} \\left(- \\cos{\\left (2 \\theta \\right )} + 1\\right)^{2} \\sin{\\left (2 \\phi \\right )} - r^{2} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} - \\sqrt{2} r^{2} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\sin{\\left (\\phi + \\frac{\\pi}{4} \\right )} \\cos{\\left (\\theta \\right )} + r^{2} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} - \\cos{\\left (\\theta \\right )}\\right) \\right) \\mathrm{d} x\\wedge \\mathrm{d} z + \\left( \\frac{r^{2} \\sin{\\left (\\theta \\right )}}{\\sqrt{r^{2} + 1}} \\left(r^{2} \\sin^{3}{\\left (\\theta \\right )} \\cos^{2}{\\left (\\phi \\right )} + \\sqrt{2} r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} \\cos{\\left (\\phi + \\frac{\\pi}{4} \\right )} + \\frac{r}{2} \\sin{\\left (2 \\theta \\right )} - r \\cos{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )}\\right) \\right) \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "*omega = r*(r**3*sin(theta)**3*cos(phi)*cos(theta) - sqrt(2)*r**2*sin(theta)**3*sin(phi + pi/4) + sqrt(2)*r**2*sin(theta)*sin(phi + pi/4) - r**2*cos(theta)**3 + sin(theta)*cos(phi) - cos(theta))/sqrt(r**2 + 1) dx/\\dy + r*(-r**3*(-cos(2*theta) + 1)**2*sin(2*phi)/8 - r**2*sin(phi)*sin(theta)**3 - sqrt(2)*r**2*sin(phi)*sin(theta)**2*sin(phi + pi/4)*cos(theta) + r**2*sin(phi)*sin(theta) - cos(theta))/sqrt(r**2 + 1) dx/\\dz + r**2*(r**2*sin(theta)**3*cos(phi)**2 + sqrt(2)*r*sin(phi)*sin(theta)*cos(theta)*cos(phi + pi/4) + r*sin(2*theta)/2 - r*cos(phi)*cos(theta)**2 + sin(theta))*sin(theta)/sqrt(r**2 + 1) dy/\\dz"
      ]
     },
     "execution_count": 266,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "som = om.hodge_dual(g)\n",
    "print(som) ; som.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>The Hodge dual of a 2-form is a 1-form:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 267,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2-form A on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    }
   ],
   "source": [
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 268,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1-form *A on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\star A = \\left( \\frac{r^{2}}{\\sqrt{r^{2} + 1}} \\left(3 r^{4} \\sin^{3}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} + 3 r^{4} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} - 2 r^{3} \\sin^{3}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + 3 r^{3} \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\theta \\right )} - 2 r^{3} \\sin{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + 3 r^{3} \\cos^{5}{\\left (\\theta \\right )} - 2 r^{2} \\sin^{3}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 2 r^{2} \\sin^{3}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + 2 r^{2} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos^{3}{\\left (\\theta \\right )} - \\frac{r^{2}}{8} \\sin{\\left (2 \\phi \\right )} + \\frac{r^{2}}{16} \\sin{\\left (2 \\phi - 4 \\theta \\right )} + \\frac{r^{2}}{16} \\sin{\\left (2 \\phi + 4 \\theta \\right )} - r \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos^{2}{\\left (\\phi \\right )} + r \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + r \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{2}{\\left (\\theta \\right )} + 3 r \\cos^{3}{\\left (\\theta \\right )} - 2 \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\frac{1}{2} \\cos{\\left (\\phi - 2 \\theta \\right )} - \\frac{1}{2} \\cos{\\left (\\phi + 2 \\theta \\right )}\\right) \\right) \\mathrm{d} x + \\left( \\frac{r}{\\sqrt{r^{2} + 1}} \\left(- 3 r^{5} \\sin^{2}{\\left (\\phi \\right )} \\sin^{6}{\\left (\\theta \\right )} + 3 r^{5} \\sin^{2}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} + 3 r^{5} \\sin^{4}{\\left (\\theta \\right )} - 3 r^{5} \\sin^{2}{\\left (\\theta \\right )} - 2 r^{4} \\sin^{2}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - 3 r^{4} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos^{3}{\\left (\\theta \\right )} - 2 r^{3} \\sin^{4}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} + 2 r^{3} \\sin^{2}{\\left (\\phi \\right )} \\sin^{4}{\\left (\\theta \\right )} - 2 r^{3} \\sin^{2}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + 3 r^{3} \\sin^{4}{\\left (\\theta \\right )} - 3 r^{3} \\sin^{2}{\\left (\\theta \\right )} + r^{2} \\sin^{2}{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - \\sqrt{2} r^{2} \\sin^{3}{\\left (\\theta \\right )} \\sin{\\left (\\phi + \\frac{\\pi}{4} \\right )} + r^{2} \\sin^{3}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\phi \\right )} - r^{2} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + \\sqrt{2} r^{2} \\sin{\\left (\\theta \\right )} \\sin{\\left (\\phi + \\frac{\\pi}{4} \\right )} - r^{2} \\cos^{3}{\\left (\\theta \\right )} + \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - \\cos{\\left (\\theta \\right )}\\right) \\right) \\mathrm{d} y + \\left( \\frac{r}{\\sqrt{r^{2} + 1}} \\left(3 r^{5} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos^{3}{\\left (\\theta \\right )} + 2 r^{4} \\sin{\\left (\\phi \\right )} \\sin^{5}{\\left (\\theta \\right )} - 3 r^{4} \\sin^{5}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + 6 r^{4} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} - 3 r^{4} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + \\frac{r^{3}}{4} \\left(- \\cos{\\left (2 \\theta \\right )} + 1\\right)^{2} \\sin{\\left (2 \\phi \\right )} - 2 r^{3} \\sin^{3}{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} + 2 r^{3} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - 2 r^{3} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} + r^{2} \\sin{\\left (\\phi \\right )} \\sin^{3}{\\left (\\theta \\right )} - r^{2} \\sin{\\left (\\phi \\right )} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} + r^{2} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} - r^{2} \\sin^{2}{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )} - \\cos{\\left (\\theta \\right )}\\right) \\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "*A = r**2*(3*r**4*sin(phi)**3*sin(theta)**4*cos(phi)*cos(theta)**2 + 3*r**4*sin(phi)*sin(theta)**4*cos(phi)**3*cos(theta)**2 - 2*r**3*sin(phi)**3*sin(theta)**4*cos(phi)*cos(theta) + 3*r**3*sin(phi)**2*sin(theta)**2*cos(theta)**3 - 2*r**3*sin(phi)*sin(theta)**4*cos(phi)**3*cos(theta) + 3*r**3*cos(theta)**5 - 2*r**2*sin(phi)**3*sin(theta)**4*cos(phi) + 2*r**2*sin(phi)**3*sin(theta)**3*cos(theta) + 2*r**2*sin(phi)*sin(theta)*cos(theta)**3 - r**2*sin(2*phi)/8 + r**2*sin(2*phi - 4*theta)/16 + r**2*sin(2*phi + 4*theta)/16 - r*sin(phi)*sin(theta)**3*cos(phi)**2 + r*sin(phi)*sin(theta)**2*cos(phi)*cos(theta) + r*sin(theta)*cos(phi)*cos(theta)**2 + 3*r*cos(theta)**3 - 2*sin(phi)*sin(theta)**2*cos(phi) + cos(phi - 2*theta)/2 - cos(phi + 2*theta)/2)/sqrt(r**2 + 1) dx + r*(-3*r**5*sin(phi)**2*sin(theta)**6 + 3*r**5*sin(phi)**2*sin(theta)**4 + 3*r**5*sin(theta)**4 - 3*r**5*sin(theta)**2 - 2*r**4*sin(phi)**2*sin(theta)**4*cos(theta) - 3*r**4*sin(phi)*sin(theta)**2*cos(phi)*cos(theta)**3 - 2*r**3*sin(phi)**4*sin(theta)**4 + 2*r**3*sin(phi)**2*sin(theta)**4 - 2*r**3*sin(phi)**2*sin(theta)**3*cos(phi)*cos(theta) + 3*r**3*sin(theta)**4 - 3*r**3*sin(theta)**2 + r**2*sin(phi)**2*sin(theta)**2*cos(theta) - sqrt(2)*r**2*sin(theta)**3*sin(phi + pi/4) + r**2*sin(theta)**3*cos(phi)**3 - r**2*sin(theta)**2*cos(theta) + sqrt(2)*r**2*sin(theta)*sin(phi + pi/4) - r**2*cos(theta)**3 + sin(theta)*cos(phi) - cos(theta))/sqrt(r**2 + 1) dy + r*(3*r**5*sin(phi)*sin(theta)**3*cos(theta)**3 + 2*r**4*sin(phi)*sin(theta)**5 - 3*r**4*sin(theta)**5*cos(phi) + 6*r**4*sin(theta)**3*cos(phi) - 3*r**4*sin(theta)*cos(phi) + r**3*(-cos(2*theta) + 1)**2*sin(2*phi)/4 - 2*r**3*sin(phi)**3*sin(theta)**3*cos(theta) + 2*r**3*sin(phi)*sin(theta)**3*cos(theta) - 2*r**3*sin(phi)*sin(theta)**2*cos(phi) + r**2*sin(phi)*sin(theta)**3 - r**2*sin(phi)*sin(theta)**2*cos(phi)*cos(theta) + r**2*sin(phi)*sin(theta) - r**2*sin(theta)**2*cos(theta) - cos(theta))/sqrt(r**2 + 1) dz"
      ]
     },
     "execution_count": 268,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sa = a.hodge_dual(g)\n",
    "print(sa) ; sa.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Finally, the Hodge dual of a 3-form is a 0-form:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 269,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3-form dA on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{d}A = \\left( - 6 r^{3} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos^{2}{\\left (\\theta \\right )} - 2 r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} - 1 \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "dA = (-6*r**3*sin(phi)*sin(theta)*cos(theta)**2 - 2*r*sin(phi)*sin(theta) - 1) dx/\\dy/\\dz"
      ]
     },
     "execution_count": 269,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(da) ; da.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 270,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scalar field *dA on the Open subset U of the 3-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\star \\mathrm{d}A:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & - \\frac{1}{6} \\left(36 y z^{2} + 12 y + 6\\right) \\sqrt{x^{2} + y^{2} + z^{2} + 1} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & - \\frac{1}{6} \\sqrt{r^{2} + 1} \\left(36 r^{3} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} \\cos^{2}{\\left (\\theta \\right )} + 12 r \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} + 6\\right) \\end{array}</script></html>"
      ],
      "text/plain": [
       "*dA: U --> R\n",
       "   (x, y, z) |--> -(36*y*z**2 + 12*y + 6)*sqrt(x**2 + y**2 + z**2 + 1)/6\n",
       "   (r, theta, phi) |--> -sqrt(r**2 + 1)*(36*r**3*sin(phi)*sin(theta)*cos(theta)**2 + 12*r*sin(phi)*sin(theta) + 6)/6"
      ]
     },
     "execution_count": 270,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sda = da.hodge_dual(g)\n",
    "print(sda) ; sda.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>In dimension 3 and for a Riemannian metric, the Hodge star is idempotent:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 271,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 271,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sf.hodge_dual(g) == f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 272,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
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       "True"
      ]
     },
     "execution_count": 272,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "som.hodge_dual(g) == om"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 273,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 273,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sa.hodge_dual(g) == a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 274,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 274,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sda.hodge_dual(g) == da"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Getting help</h2>\n",
    "<p><span>To get the list of functions (methods) that can be called on a object, type the name of the object, followed by a dot and the TAB key, e.g. </span></p>\n",
    "<p><span><span style=\"font-family: courier new,courier;\">sa.<tab></span><br /></span></p>\n",
    "<p><span id=\"cell_outer_269\">To get information on an object or a method, use the question mark:</span></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 275,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "nabla?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 276,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "g.ricci_scalar?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<p>Using a double question mark leads directly to the <strong>Python</strong> <strong>source code</strong> (SageMath is <span style=\"color: #ff0000;\"><strong>open source</strong></span>, isn't it?)<span style=\"color: #ff0000;\"><br /></span></p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 277,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "g.ricci_scalar??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<h2>Going further</h2>\n",
    "<p>Have a look at the <a href=\"http://sagemanifolds.obspm.fr/examples.html\">examples on SageManifolds page</a>, especially the <a href=\"http://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/v1.0/SM_sphere_S2.ipynb\">2-dimensional sphere example</a> for usage on a non-parallelizable manifold (each scalar field has to be defined in at least two coordinate charts, the module $\\mathfrak{X}(\\mathcal{M})$ is no longer free and each tensor field has to be defined in at least two vector frames).</p>"
   ]
  }
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