Worksheets/vector_calc_change.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vector calculus with SageMath\n",
    "\n",
    "## 4. Changing coordinates\n",
    "\n",
    "This notebook illustrates operations on vector fields on Euclidean spaces, as introduced in Trac ticket [#24623](https://trac.sagemath.org/ticket/24623)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'SageMath version 8.2.rc4, Release Date: 2018-04-20'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%display latex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cartesian coordinates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by declaring the 3-dimensional Euclidean space $\\mathbb{E}^3$, with $(x,y,z)$ as Cartesian coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Euclidean space E^3\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbb{E}^{3}</script></html>"
      ],
      "text/plain": [
       "Euclidean space E^3"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.<x,y,z> = EuclideanSpace()\n",
    "print(E)\n",
    "E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{E}^3$ is endowed with the chart of Cartesian coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3},(x, y, z)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Chart (E^3, (x, y, z))]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.atlas()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us denote by `cartesian` the chart of Cartesian coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3},(x, y, z)\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (E^3, (x, y, z))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian = E.cartesian_coordinates()\n",
    "cartesian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The access to the individual coordinates is performed via the square bracket operator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x</script></html>"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "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": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thanks to use of `<x,y,z>` when declaring `E`, the Python variables `x`, `y` and `z` have been created (i.e. there is no need to declare them by something like `y = var('y')`); they represent the coordinates $(x,y,z)$ as symbolic expressions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "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": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y is cartesian[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "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": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each of the Cartesian coordinates spans the entire real line:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x :\\ \\left( -\\infty, +\\infty \\right) ;\\quad y :\\ \\left( -\\infty, +\\infty \\right) ;\\quad z :\\ \\left( -\\infty, +\\infty \\right)</script></html>"
      ],
      "text/plain": [
       "x: (-oo, +oo); y: (-oo, +oo); z: (-oo, +oo)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian.coord_range()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Being the only coordinate chart created so far, `cartesian` is the default chart on `E`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "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": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian is E.default_chart()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{E}^3$ is endowed with the orthonormal vector frame $(e_x, e_y, e_z)$ associated with Cartesian coordinates: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3}, \\left(e_{ x },e_{ y },e_{ z }\\right)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Coordinate frame (E^3, (e_x,e_y,e_z))]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.frames()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us denote it by `cartesian_frame`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3}, \\left(e_{ x },e_{ y },e_{ z }\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "Coordinate frame (E^3, (e_x,e_y,e_z))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian_frame = E.cartesian_frame()\n",
    "cartesian_frame"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "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": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian_frame is E.default_frame()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each element of this frame is a unit vector field; for instance, we have $e_x\\cdot e_x = 1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ x }</script></html>"
      ],
      "text/plain": [
       "Vector field e_x on the Euclidean space E^3"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_x = cartesian_frame[1]\n",
    "e_x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}1</script></html>"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_x.dot(e_x).expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "as well as $e_x\\cdot e_y = 0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_y = cartesian_frame[2]\n",
    "e_x.dot(e_y).expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Spherical coordinates\n",
    "\n",
    "Spherical coordinates are introduced by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3},(r, {\\theta}, {\\phi})\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (E^3, (r, th, ph))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spherical.<r,th,ph> = E.spherical_coordinates()\n",
    "spherical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(r, {\\theta}, {\\phi}\\right)</script></html>"
      ],
      "text/plain": [
       "(r, th, ph)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spherical[:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\theta} :\\ \\left( 0 , \\pi \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right)</script></html>"
      ],
      "text/plain": [
       "r: (0, +oo); th: (0, pi); ph: (0, 2*pi)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spherical.coord_range()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{E}^3$ is now endowed with two coordinate charts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3},(x, y, z)\\right), \\left(\\mathbb{E}^{3},(r, {\\theta}, {\\phi})\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Chart (E^3, (x, y, z)), Chart (E^3, (r, th, ph))]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.atlas()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The change-of-coordinate formulas have been automatically implemented during the above call `E.spherical_coordinates()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "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(ph)*sin(th)\n",
       "y = r*sin(ph)*sin(th)\n",
       "z = r*cos(th)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.coord_change(spherical, cartesian).display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "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",
       "th = arctan2(sqrt(x^2 + y^2), z)\n",
       "ph = arctan2(y, x)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.coord_change(cartesian, spherical).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{E}^3$ is also now endowed with three vector frames:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3}, \\left(e_{ x },e_{ y },e_{ z }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(e_{ r },e_{ {\\theta} },e_{ {\\phi} }\\right)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Coordinate frame (E^3, (e_x,e_y,e_z)),\n",
       " Coordinate frame (E^3, (d/dr,d/dth,d/dph)),\n",
       " Vector frame (E^3, (e_r,e_th,e_ph))]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.frames()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second one is the coordinate frame of spherical coordinates, while the third one is the standard orthonormal frame associated with spherical coordinates. For Cartesian coordinates, the coordinate frame and the orthonormal frame coincide: it is $(e_x,e_y,e_z)$. For spherical coordinates, the orthonormal frame is denoted $(e_r,e_\\theta,e_\\phi)$ and is returned by the method `spherical_frame()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3}, \\left(e_{ r },e_{ {\\theta} },e_{ {\\phi} }\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "Vector frame (E^3, (e_r,e_th,e_ph))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spherical_frame = E.spherical_frame()\n",
    "spherical_frame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We may check that it is an orthonormal frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[1, 0, 0\\right], \\left[0, 1, 0\\right], \\left[0, 0, 1\\right]\\right]</script></html>"
      ],
      "text/plain": [
       "[[1, 0, 0], [0, 1, 0], [0, 0, 1]]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "es = spherical_frame\n",
    "[[es[i].dot(es[j]).expr() for j in E.irange()] for i in E.irange()]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orthonormal spherical frame expressed in terms of the Cartesian one: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ r } = \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) e_{ x } + \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) e_{ y } + \\cos\\left({\\theta}\\right) e_{ z }</script></html>"
      ],
      "text/plain": [
       "e_r = cos(ph)*sin(th) e_x + sin(ph)*sin(th) e_y + cos(th) e_z"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\theta} } = \\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) e_{ x } + \\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) e_{ y } -\\sin\\left({\\theta}\\right) e_{ z }</script></html>"
      ],
      "text/plain": [
       "e_th = cos(ph)*cos(th) e_x + cos(th)*sin(ph) e_y - sin(th) e_z"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\phi} } = -\\sin\\left({\\phi}\\right) e_{ x } + \\cos\\left({\\phi}\\right) e_{ y }</script></html>"
      ],
      "text/plain": [
       "e_ph = -sin(ph) e_x + cos(ph) e_y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in spherical_frame:\n",
    "    show(vec.display(cartesian_frame, spherical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The reverse transformation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ x } = \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) e_{ r } + \\cos\\left({\\phi}\\right) \\cos\\left({\\theta}\\right) e_{ {\\theta} } -\\sin\\left({\\phi}\\right) e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_x = cos(ph)*sin(th) e_r + cos(ph)*cos(th) e_th - sin(ph) e_ph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ y } = \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) e_{ r } + \\cos\\left({\\theta}\\right) \\sin\\left({\\phi}\\right) e_{ {\\theta} } + \\cos\\left({\\phi}\\right) e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_y = sin(ph)*sin(th) e_r + cos(th)*sin(ph) e_th + cos(ph) e_ph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ z } = \\cos\\left({\\theta}\\right) e_{ r } -\\sin\\left({\\theta}\\right) e_{ {\\theta} }</script></html>"
      ],
      "text/plain": [
       "e_z = cos(th) e_r - sin(th) e_th"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in cartesian_frame:\n",
    "    show(vec.display(spherical_frame, spherical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orthonormal frame $(e_r,e_\\theta,e_\\phi)$ expressed in terms on the coordinate frame $\\left(\\frac{\\partial}{\\partial r}, \\frac{\\partial}{\\partial\\theta}, \\frac{\\partial}{\\partial \\phi}\\right)$ (the latter being returned by the method `frame()` acting on the chart `spherical`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ r } = \\frac{\\partial}{\\partial r }</script></html>"
      ],
      "text/plain": [
       "e_r = d/dr"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\theta} } = \\frac{1}{r} \\frac{\\partial}{\\partial {\\theta} }</script></html>"
      ],
      "text/plain": [
       "e_th = 1/r d/dth"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\phi} } = \\frac{1}{r \\sin\\left({\\theta}\\right)} \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_ph = 1/(r*sin(th)) d/dph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in spherical_frame:\n",
    "    show(vec.display(spherical.frame(), spherical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cylindrical coordinates\n",
    "\n",
    "Cylindrical coordinates are introduced by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3},({\\rho}, {\\phi}, z)\\right)</script></html>"
      ],
      "text/plain": [
       "Chart (E^3, (rh, ph, z))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cylindrical.<rh,ph,z> = E.cylindrical_coordinates()\n",
    "cylindrical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left({\\rho}, {\\phi}, z\\right)</script></html>"
      ],
      "text/plain": [
       "(rh, ph, z)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cylindrical[:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "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": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rh is cylindrical[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\rho} :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\phi} :\\ \\left( 0 , 2 \\, \\pi \\right) ;\\quad z :\\ \\left( -\\infty, +\\infty \\right)</script></html>"
      ],
      "text/plain": [
       "rh: (0, +oo); ph: (0, 2*pi); z: (-oo, +oo)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cylindrical.coord_range()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{E}^3$ is now endowed with two coordinate charts:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3},(x, y, z)\\right), \\left(\\mathbb{E}^{3},(r, {\\theta}, {\\phi})\\right), \\left(\\mathbb{E}^{3},({\\rho}, {\\phi}, z)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Chart (E^3, (x, y, z)), Chart (E^3, (r, th, ph)), Chart (E^3, (rh, ph, z))]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.atlas()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transformations linking the cylindrical coordinates to the Cartesian ones are"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & {\\rho} \\cos\\left({\\phi}\\right) \\\\ y & = & {\\rho} \\sin\\left({\\phi}\\right) \\\\ z & = & z \\end{array}\\right.</script></html>"
      ],
      "text/plain": [
       "x = rh*cos(ph)\n",
       "y = rh*sin(ph)\n",
       "z = z"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.coord_change(cylindrical, cartesian).display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\rho} & = & \\sqrt{x^{2} + y^{2}} \\\\ {\\phi} & = & \\arctan\\left(y, x\\right) \\\\ z & = & z \\end{array}\\right.</script></html>"
      ],
      "text/plain": [
       "rh = sqrt(x^2 + y^2)\n",
       "ph = arctan2(y, x)\n",
       "z = z"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.coord_change(cartesian, cylindrical).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{E}^3$ is also now endowed with five vector frames:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{E}^{3}, \\left(e_{ x },e_{ y },e_{ z }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(\\frac{\\partial}{\\partial r },\\frac{\\partial}{\\partial {\\theta} },\\frac{\\partial}{\\partial {\\phi} }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(e_{ r },e_{ {\\theta} },e_{ {\\phi} }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(\\frac{\\partial}{\\partial {\\rho} },\\frac{\\partial}{\\partial {\\phi} },\\frac{\\partial}{\\partial z }\\right)\\right), \\left(\\mathbb{E}^{3}, \\left(e_{ {\\rho} },e_{ {\\phi} },e_{ z }\\right)\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[Coordinate frame (E^3, (e_x,e_y,e_z)),\n",
       " Coordinate frame (E^3, (d/dr,d/dth,d/dph)),\n",
       " Vector frame (E^3, (e_r,e_th,e_ph)),\n",
       " Coordinate frame (E^3, (d/drh,d/dph,d/dz)),\n",
       " Vector frame (E^3, (e_rh,e_ph,e_z))]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.frames()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orthonormal frame associated with cylindrical coordinates is $(e_\\rho, e_\\phi, e_z)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{3}, \\left(e_{ {\\rho} },e_{ {\\phi} },e_{ z }\\right)\\right)</script></html>"
      ],
      "text/plain": [
       "Vector frame (E^3, (e_rh,e_ph,e_z))"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cylindrical_frame = E.cylindrical_frame()\n",
    "cylindrical_frame"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We may check that it is an orthonormal frame:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[1, 0, 0\\right], \\left[0, 1, 0\\right], \\left[0, 0, 1\\right]\\right]</script></html>"
      ],
      "text/plain": [
       "[[1, 0, 0], [0, 1, 0], [0, 0, 1]]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ec = cylindrical_frame\n",
    "[[ec[i].dot(ec[j]).expr() for j in E.irange()] for i in E.irange()]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orthonormal cylindrical frame expressed in terms of the Cartesian one: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\rho} } = \\cos\\left({\\phi}\\right) e_{ x } + \\sin\\left({\\phi}\\right) e_{ y }</script></html>"
      ],
      "text/plain": [
       "e_rh = cos(ph) e_x + sin(ph) e_y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\phi} } = -\\sin\\left({\\phi}\\right) e_{ x } + \\cos\\left({\\phi}\\right) e_{ y }</script></html>"
      ],
      "text/plain": [
       "e_ph = -sin(ph) e_x + cos(ph) e_y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ z } = e_{ z }</script></html>"
      ],
      "text/plain": [
       "e_z = e_z"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in cylindrical_frame:\n",
    "    show(vec.display(cartesian_frame, cylindrical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The reverse transformation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ x } = \\cos\\left({\\phi}\\right) e_{ {\\rho} } -\\sin\\left({\\phi}\\right) e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_x = cos(ph) e_rh - sin(ph) e_ph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ y } = \\sin\\left({\\phi}\\right) e_{ {\\rho} } + \\cos\\left({\\phi}\\right) e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_y = sin(ph) e_rh + cos(ph) e_ph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ z } = e_{ z }</script></html>"
      ],
      "text/plain": [
       "e_z = e_z"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in cartesian_frame:\n",
    "    show(vec.display(cylindrical_frame, cylindrical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orthonormal cylindrical frame expressed in terms of the spherical one: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\rho} } = \\sin\\left({\\theta}\\right) e_{ r } + \\cos\\left({\\theta}\\right) e_{ {\\theta} }</script></html>"
      ],
      "text/plain": [
       "e_rh = sin(th) e_r + cos(th) e_th"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\phi} } = e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_ph = e_ph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ z } = \\cos\\left({\\theta}\\right) e_{ r } -\\sin\\left({\\theta}\\right) e_{ {\\theta} }</script></html>"
      ],
      "text/plain": [
       "e_z = cos(th) e_r - sin(th) e_th"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in cylindrical_frame:\n",
    "    show(vec.display(spherical_frame, spherical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The reverse transformation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ r } = \\sin\\left({\\theta}\\right) e_{ {\\rho} } + \\cos\\left({\\theta}\\right) e_{ z }</script></html>"
      ],
      "text/plain": [
       "e_r = sin(th) e_rh + cos(th) e_z"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\theta} } = \\cos\\left({\\theta}\\right) e_{ {\\rho} } -\\sin\\left({\\theta}\\right) e_{ z }</script></html>"
      ],
      "text/plain": [
       "e_th = cos(th) e_rh - sin(th) e_z"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\phi} } = e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_ph = e_ph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in spherical_frame:\n",
    "    show(vec.display(cylindrical_frame, spherical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orthonormal frame $(e_\\rho,e_\\phi,e_z)$ expressed in terms on the coordinate frame $\\left(\\frac{\\partial}{\\partial\\rho}, \\frac{\\partial}{\\partial\\phi}, \\frac{\\partial}{\\partial z}\\right)$ (the latter being returned by the method `frame()` acting on the chart `cylindrical`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\rho} } = \\frac{\\partial}{\\partial {\\rho} }</script></html>"
      ],
      "text/plain": [
       "e_rh = d/drh"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ {\\phi} } = \\frac{1}{{\\rho}} \\frac{\\partial}{\\partial {\\phi} }</script></html>"
      ],
      "text/plain": [
       "e_ph = 1/rh d/dph"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}e_{ z } = \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "e_z = d/dz"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for vec in cylindrical_frame:\n",
    "    show(vec.display(cylindrical.frame(), cylindrical))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coordinates of a point\n",
    "\n",
    "We introduce a point $p\\in \\mathbb{E}^3$ via the generic SageMath syntax for creating an element from its parent (here $\\mathbb{E}^3$), i.e. the call operator `()`, with the coordinates of the point as the first argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Point p on the Euclidean space E^3\n"
     ]
    }
   ],
   "source": [
    "p = E((-1, 1,0), chart=cartesian, name='p')\n",
    "print(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Actually, since the Cartesian coordinates are the default ones, the above writting is equivalent to"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Point p on the Euclidean space E^3\n"
     ]
    }
   ],
   "source": [
    "p = E((-1, 1,0), name='p')\n",
    "print(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The coordinates of $p$ in a given coordinate chart are obtained by letting the corresponding chart act on $p$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-1, 1, 0\\right)</script></html>"
      ],
      "text/plain": [
       "(-1, 1, 0)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\sqrt{2}, \\frac{1}{2} \\, \\pi, \\frac{3}{4} \\, \\pi\\right)</script></html>"
      ],
      "text/plain": [
       "(sqrt(2), 1/2*pi, 3/4*pi)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spherical(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\sqrt{2}, \\frac{3}{4} \\, \\pi, 0\\right)</script></html>"
      ],
      "text/plain": [
       "(sqrt(2), 3/4*pi, 0)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cylindrical(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An example of point defined from its spherical coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Point q on the Euclidean space E^3\n"
     ]
    }
   ],
   "source": [
    "q = E((4,pi/3,pi), chart=spherical, name='q')\n",
    "print(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(4, \\frac{1}{3} \\, \\pi, \\pi\\right)</script></html>"
      ],
      "text/plain": [
       "(4, 1/3*pi, pi)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spherical(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-2 \\, \\sqrt{3}, 0, 2\\right)</script></html>"
      ],
      "text/plain": [
       "(-2*sqrt(3), 0, 2)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cartesian(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(2 \\, \\sqrt{3}, \\pi, 2\\right)</script></html>"
      ],
      "text/plain": [
       "(2*sqrt(3), pi, 2)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cylindrical(q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expressions of a scalar field in various coordinate systems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us define a scalar field on $\\mathbb{E}^3$ from its expression in Cartesian coordinates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = E.scalar_field(x^2+y^2 - z^2, name='f')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that since the Cartesian coordinates are the default ones, we did not specify them in the above definition. Thanks to the known coordinate transformations, the expression of $f$ in terms of other coordinates is automatically computed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x^{2} + y^{2} - z^{2} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & 2 \\, r^{2} \\sin\\left({\\theta}\\right)^{2} - r^{2} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & {\\rho}^{2} - z^{2} \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: E^3 --> R\n",
       "   (x, y, z) |--> x^2 + y^2 - z^2\n",
       "   (r, th, ph) |--> 2*r^2*sin(th)^2 - r^2\n",
       "   (rh, ph, z) |--> rh^2 - z^2"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can limit the output to a single coordinate system:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x^{2} + y^{2} - z^{2} \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: E^3 --> R\n",
       "   (x, y, z) |--> x^2 + y^2 - z^2"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display(cartesian)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} f:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & {\\rho}^{2} - z^{2} \\end{array}</script></html>"
      ],
      "text/plain": [
       "f: E^3 --> R\n",
       "   (rh, ph, z) |--> rh^2 - z^2"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.display(cylindrical)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The coordinate expression in a given coordinate system is obtained via the method `expr()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{2} + y^{2} - z^{2}</script></html>"
      ],
      "text/plain": [
       "x^2 + y^2 - z^2"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expr()  # expression in the default chart (Cartesian coordinates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, r^{2} \\sin\\left({\\theta}\\right)^{2} - r^{2}</script></html>"
      ],
      "text/plain": [
       "2*r^2*sin(th)^2 - r^2"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expr(spherical)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\rho}^{2} - z^{2}</script></html>"
      ],
      "text/plain": [
       "rh^2 - z^2"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expr(cylindrical)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value of $f$ at points $p$ and $q$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2</script></html>"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}8</script></html>"
      ],
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, we may define a scalar field from its coordinate expression in a chart that is not the default one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = E.scalar_field(r^2, chart=spherical, name='g')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of using the keyword argument `chart`, one can pass a dictionary as the first argument, with the chart as key:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = E.scalar_field({spherical: r^2}, name='g')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} g:& \\mathbb{E}^{3} & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & x^{2} + y^{2} + z^{2} \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{2} \\\\ & \\left({\\rho}, {\\phi}, z\\right) & \\longmapsto & {\\rho}^{2} + z^{2} \\end{array}</script></html>"
      ],
      "text/plain": [
       "g: E^3 --> R\n",
       "   (x, y, z) |--> x^2 + y^2 + z^2\n",
       "   (r, th, ph) |--> r^2\n",
       "   (rh, ph, z) |--> rh^2 + z^2"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expression of a vector field in various frames\n",
    "\n",
    "Let us introduce a vector field on $\\mathbb{E}^3$ by its components in the Cartesian frame. Since the latter is the default vector frame on $\\mathbb{E}^3$, it suffices to write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = -y e_{ x } + x e_{ y } + z^{2} e_{ z }</script></html>"
      ],
      "text/plain": [
       "v = -y e_x + x e_y + z^2 e_z"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = E.vector_field(-y, x, z^2, name='v')\n",
    "v.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Equivalently, a vector field can be defined directly from its expansion on the Cartesian frame: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-y e_{ x } + x e_{ y } + z^{2} e_{ z }</script></html>"
      ],
      "text/plain": [
       "-y e_x + x e_y + z^2 e_z"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex, ey, ez = cartesian_frame[:]\n",
    "v = -y*ex + x*ey + z^2*ez\n",
    "v.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us provide `v` with some name, as above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = -y e_{ x } + x e_{ y } + z^{2} e_{ z }</script></html>"
      ],
      "text/plain": [
       "v = -y e_x + x e_y + z^2 e_z"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.set_name('v')\n",
    "v.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The components of $v$ are returned by the square bracket operator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-y</script></html>"
      ],
      "text/plain": [
       "-y"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-y, x, z^{2}\\right]</script></html>"
      ],
      "text/plain": [
       "[-y, x, z^2]"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expression of $v$ in terms of the orthonormal spherical frame is obtained by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\left( \\frac{z^{3}}{\\sqrt{x^{2} + y^{2} + z^{2}}} \\right) e_{ r } + \\left( -\\frac{\\sqrt{x^{2} + y^{2}} z^{2}}{\\sqrt{x^{2} + y^{2} + z^{2}}} \\right) e_{ {\\theta} } + \\left( \\sqrt{x^{2} + y^{2}} \\right) e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = z^3/sqrt(x^2 + y^2 + z^2) e_r - sqrt(x^2 + y^2)*z^2/sqrt(x^2 + y^2 + z^2) e_th + sqrt(x^2 + y^2) e_ph"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display(spherical_frame)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We note that the components are still expressed in the default chart (Cartesian coordinates). To have them expressed in the spherical chart, it suffices to pass the latter as a second argument to `display()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = r^{2} \\cos\\left({\\theta}\\right)^{3} e_{ r } -r^{2} \\cos\\left({\\theta}\\right)^{2} \\sin\\left({\\theta}\\right) e_{ {\\theta} } + r \\sin\\left({\\theta}\\right) e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = r^2*cos(th)^3 e_r - r^2*cos(th)^2*sin(th) e_th + r*sin(th) e_ph"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display(spherical_frame, spherical)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, the components of $v$ are obtained by means of the square bracket operator, by specify the vector frame as first argument, and the coordinate chart as the last one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{z^{3}}{\\sqrt{x^{2} + y^{2} + z^{2}}}</script></html>"
      ],
      "text/plain": [
       "z^3/sqrt(x^2 + y^2 + z^2)"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[spherical_frame,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}r^{2} \\cos\\left({\\theta}\\right)^{3}</script></html>"
      ],
      "text/plain": [
       "r^2*cos(th)^3"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[spherical_frame,1,spherical]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[r^{2} \\cos\\left({\\theta}\\right)^{3}, -r^{2} \\cos\\left({\\theta}\\right)^{2} \\sin\\left({\\theta}\\right), r \\sin\\left({\\theta}\\right)\\right]</script></html>"
      ],
      "text/plain": [
       "[r^2*cos(th)^3, -r^2*cos(th)^2*sin(th), r*sin(th)]"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[spherical_frame,:,spherical]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, the expression of $v$ in terms of the cylindrical frame is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = {\\rho} e_{ {\\phi} } + z^{2} e_{ z }</script></html>"
      ],
      "text/plain": [
       "v = rh e_ph + z^2 e_z"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display(cylindrical_frame, cylindrical)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0, {\\rho}, z^{2}\\right]</script></html>"
      ],
      "text/plain": [
       "[0, rh, z^2]"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[cylindrical_frame,:,cylindrical]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value of the vector field $v$ at point $p$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vector v at Point p on the Euclidean space E^3\n"
     ]
    }
   ],
   "source": [
    "vp = v.at(p)\n",
    "print(vp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = -e_{ x }-e_{ y }</script></html>"
      ],
      "text/plain": [
       "v = -e_x - e_y"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vp.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\sqrt{2} e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = sqrt(2) e_ph"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vp.display(spherical_frame.at(p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = \\sqrt{2} e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = sqrt(2) e_ph"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vp.display(cylindrical_frame.at(p))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value of the vector field $v$ at point $q$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vector v at Point q on the Euclidean space E^3\n"
     ]
    }
   ],
   "source": [
    "vq = v.at(q)\n",
    "print(vq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = -2 \\, \\sqrt{3} e_{ y } + 4 e_{ z }</script></html>"
      ],
      "text/plain": [
       "v = -2*sqrt(3) e_y + 4 e_z"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vq.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = 2 e_{ r } -2 \\, \\sqrt{3} e_{ {\\theta} } + 2 \\, \\sqrt{3} e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = 2 e_r - 2*sqrt(3) e_th + 2*sqrt(3) e_ph"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vq.display(spherical_frame.at(q))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = 2 \\, \\sqrt{3} e_{ {\\phi} } + 4 e_{ z }</script></html>"
      ],
      "text/plain": [
       "v = 2*sqrt(3) e_ph + 4 e_z"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vq.display(cylindrical_frame.at(q))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Changing the default coordinates and default vector frame\n",
    "\n",
    "At any time, one may change the default coordinates by the method `set_default_chart`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "E.set_default_chart(spherical)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, r^{2} \\sin\\left({\\theta}\\right)^{2} - r^{2}</script></html>"
      ],
      "text/plain": [
       "2*r^2*sin(th)^2 - r^2"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.expr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = -r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) e_{ x } + r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) e_{ y } + r^{2} \\cos\\left({\\theta}\\right)^{2} e_{ z }</script></html>"
      ],
      "text/plain": [
       "v = -r*sin(ph)*sin(th) e_x + r*cos(ph)*sin(th) e_y + r^2*cos(th)^2 e_z"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the default vector frame is still the Cartesian one; to change to the orthonormal spherical frame, we use"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "E.set_default_frame(spherical_frame)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = r^{2} \\cos\\left({\\theta}\\right)^{3} e_{ r } -r^{2} \\cos\\left({\\theta}\\right)^{2} \\sin\\left({\\theta}\\right) e_{ {\\theta} } + r \\sin\\left({\\theta}\\right) e_{ {\\phi} }</script></html>"
      ],
      "text/plain": [
       "v = r^2*cos(th)^3 e_r - r^2*cos(th)^2*sin(th) e_th + r*sin(th) e_ph"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}v = -y e_{ x } + x e_{ y } + z^{2} e_{ z }</script></html>"
      ],
      "text/plain": [
       "v = -y e_x + x e_y + z^2 e_z"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.display(cartesian_frame, cartesian)"
   ]
  }
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