From d09eef602134df282a4997efabfdd3952857a393 Mon Sep 17 00:00:00 2001 From: AunSiro Date: Wed, 21 Mar 2018 23:45:48 +0100 Subject: [PATCH 1/2] Revision SymPy Algunos cambios menores --- notebooks_completos/040-SymPy-Intro.ipynb | 1714 ++-------- notebooks_completos/040-SymPy-Mecanica.ipynb | 2776 ----------------- notebooks_completos/041-SymPy-Mecanica.ipynb | 1808 +++++++++++ notebooks_vacios/040-SymPy-Intro.ipynb | 338 +- ...ecanica.ipynb => 041-SymPy-Mecanica.ipynb} | 259 +- 5 files changed, 2181 insertions(+), 4714 deletions(-) delete mode 100644 notebooks_completos/040-SymPy-Mecanica.ipynb create mode 100644 notebooks_completos/041-SymPy-Mecanica.ipynb rename notebooks_vacios/{040-SymPy-Mecanica.ipynb => 041-SymPy-Mecanica.ipynb} (89%) diff --git a/notebooks_completos/040-SymPy-Intro.ipynb b/notebooks_completos/040-SymPy-Intro.ipynb index 88be6c8..2a20d8b 100644 --- a/notebooks_completos/040-SymPy-Intro.ipynb +++ b/notebooks_completos/040-SymPy-Intro.ipynb @@ -11,7 +11,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# CLASE 5a: SymPy" + "# Introducción a SymPy" ] }, { @@ -20,7 +20,7 @@ "source": [ "![](http://sympy.org/static/images/logo.png)\n", "\n", - "_ __SymPy es una biblioteca de Python para matemática simbólica__. Apunta a convertirse en un sistema de algebra computacional (__CAS__) con todas sus prestaciones manteniendo el código tan simple como sea posible para manterlo comprensible y fácilmente extensible. SymPy está __escrito totalmente en Python y no requiere bibliotecas adicionales__. _Este proyecto comenzó en 2005, fue lanzado al público en 2007 y a él han contribuido durante estos años cientos de personas._\n", + "__SymPy es una biblioteca de Python para matemática simbólica__. Apunta a convertirse en un sistema de algebra computacional (__CAS__) con todas sus prestaciones manteniendo el código tan simple como sea posible para manterlo comprensible y fácilmente extensible. SymPy está __escrito totalmente en Python y no requiere bibliotecas adicionales__. _Este proyecto comenzó en 2005, fue lanzado al público en 2007 y a él han contribuido durante estos años cientos de personas._\n", "\n", "_ Otros CAS conocidos son Mathematica y Maple, sin embargo ambos son software privativo y de pago. [Aquí](https://github.com/sympy/sympy/wiki/SymPy-vs.-Maple) puedes encontrar una comparativa de SymPy con Maple. _\n", "\n", @@ -47,9 +47,7 @@ { "cell_type": "code", "execution_count": 1, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { @@ -57,7 +55,7 @@ "" ], "text/plain": [ - "" + "" ] }, "execution_count": 1, @@ -80,9 +78,7 @@ { "cell_type": "code", "execution_count": 2, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { @@ -90,7 +86,7 @@ "" ], "text/plain": [ - "" + "" ] }, "execution_count": 2, @@ -119,9 +115,7 @@ { "cell_type": "code", "execution_count": 3, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Importación\n", @@ -131,15 +125,13 @@ { "cell_type": "code", "execution_count": 4, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "IPython console for SymPy 0.7.5 (Python 3.4.2-64-bit) (ground types: python)\n", + "IPython console for SymPy 1.0 (Python 3.5.4-64-bit) (ground types: python)\n", "\n", "These commands were executed:\n", ">>> from __future__ import division\n", @@ -147,15 +139,9 @@ ">>> x, y, z, t = symbols('x y z t')\n", ">>> k, m, n = symbols('k m n', integer=True)\n", ">>> f, g, h = symbols('f g h', cls=Function)\n", + ">>> init_printing()\n", "\n", - "Documentation can be found at http://www.sympy.org\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "WARNING: Hook shutdown_hook is deprecated. Use the atexit module instead.\n" + "Documentation can be found at http://docs.sympy.org/1.0/\n" ] } ], @@ -185,17 +171,16 @@ { "cell_type": "code", "execution_count": 5, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'b' is not defined", "output_type": "error", "traceback": [ - "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 1\u001b[0m \u001b[1;31m# Intentamos usar un símbolo que no hemos creado\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0ma\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m2\u001b[0m \u001b[1;33m*\u001b[0m \u001b[0mb\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m", + "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 1\u001b[0m \u001b[1;31m# Intentamos usar un símbolo que no hemos creado\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0ma\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m2\u001b[0m \u001b[1;33m*\u001b[0m \u001b[0mb\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[1;31mNameError\u001b[0m: name 'b' is not defined" ] } @@ -217,18 +202,11 @@ { "cell_type": "code", "execution_count": 6, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\n", - "VM0GsGrEAAAAR0lEQVQIHWNgYFR2TWdgYOgU4PrAwMA+m4FnAQMDz3cGNgcGBs4FDPUBDAz1Bgz3\n", - "GQoY6i8wrOMVYOBz4E5iBqpUMRXyYAAAgLMMA5qDwz8AAAAASUVORK5CYII=\n" - ], + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAARklEQVQIHWNgYFRWCWNgYCgV4PjAwMAWycC0\ngIGB6RsDtwMDA+sChvwDDAz5Bgz3GYDUBYYVvAIMHA6c+5iBKp/YyLkzAABmaAxT1tVPwQAAAABJ\nRU5ErkJggg==\n", "text/latex": [ "$$a$$" ], @@ -250,23 +228,11 @@ { "cell_type": "code", "execution_count": 7, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAbBAMAAADc0o+gAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMmYiu80QdonvRN2Z\n", - "VKvu110NAAABaklEQVQoFY2SPUjDQBiG36R/aZv+LI5ixNWhiIuLlop7ljg5BJycUnERHUwp4iIO\n", - 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"iVBORw0KGgoAAAANSUhEUgAAAEcAAAAbBAMAAADc0o+gAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABSkAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$\\left(a + \\pi\\right)^{2}$$" ], @@ -288,21 +254,11 @@ { "cell_type": "code", "execution_count": 8, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - 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"iVBORw0KGgoAAAANSUhEUgAAADUAAAAQBAMAAABEqSrGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4UlEQVQYGWNgwARCWmuAgq8wJRgY2B4w2Bcw\nMLxEkbsC4fElMLAGoEgAOVA51g0MfH9wyHF8g8sxKquEQVRB9QE53N/YnByAdKkAxwcgBQQIuXoD\nZnaglWyRDEwLwFJIcvsYZJkaGBiYvjFwOwDlLDo6tnZ09IFVMSkwJPADBVkXMOQfYAADuJlOQO79\nBAaGfAOG+xApuJm8CgyCDH1sQLkLDCt4BVD0HWJgcGT7y8zAwOHAuQ9IgQDUTMbNxnob2D9IAt35\nxEbOHUWO4////xsYXk6ACEJJuFtQRCGcRDQxAA11NDjP+3KqAAAAAElFTkSuQmCC\n", "text/latex": [ "$$a + 2 i$$" ], @@ -323,18 +279,11 @@ { "cell_type": "code", "execution_count": 9, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAJBAMAAAASvxsjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\n", - "kgMAAAA4SURBVAgdY2BgVDYJYGBID2BVYGDtZBBZwMDdY3yQgYFvAwMQ8CmASA4gacDAfIFBBKjS\n", - "xOYAAwDWpwf9jP6jxwAAAABJRU5ErkJggg==\n" - ], + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAJBAMAAAASvxsjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\nkgMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAA4SURBVAgdY2BgVDYJYGBID2BVYGDtZBBZwMDdY3yQ\ngYFvAwMQ8CmASA4gacDAfIFBBKjSxOYAAwDWpwf9jP6jxwAAAABJRU5ErkJggg==\n", "text/latex": [ "$$e$$" ], @@ -355,9 +304,7 @@ { "cell_type": "code", "execution_count": 10, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { @@ -385,19 +332,11 @@ { "cell_type": "code", "execution_count": 11, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\n", - "VGZoascqAAAAgElEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChzFgNDvgOEvT8AzgQKrA9gEKvxCgCy\n", - "OL4zcExksAcp4jJgiHzAoAxiAgn9AIY1QBZPAgPDZAbWL0DmJQZG0e8MvB9YGVgUGHhE/zIwb7jI\n", - "IFVelMWwh+FVQgHD/v//fzGEHXfMaQAAed4eWkglV6IAAAAASUVORK5CYII=\n" - ], + "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAgUlEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA9gECo6GQBkcXxn4JjIsB+kiCuBIfIBgzKIqcTAoB/AsAbI4lFgYJjMwPoF\nyLzEwCj6nYH3AwMDiwIDj+hfBuYNFxmkyouyGPYwvEooYNj///8vhrDjjjkNAMjNHzXk+S3KAAAA\nAElFTkSuQmCC\n", "text/latex": [ "$$2 a$$" ], @@ -418,9 +357,7 @@ { "cell_type": "code", "execution_count": 12, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { @@ -449,23 +386,11 @@ { "cell_type": "code", "execution_count": 13, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - 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"iVBORw0KGgoAAAANSUhEUgAAAEEAAAAPBAMAAABJi79qAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABPklEQVQYGYWQvUoDQRSFv8WMG7MRU1uFFP5g\nM5DKyu0s3SogFgbBgDYJQbQRNi8gYiOo5AkstBK7BLERC0V8gLWwEIUggpGkWO/MxNop9hzO+WaZ\ne/FK5RA5Tv39HfGZImp2tzDKKvg/hrCqrlkQn9OsJt6Ry9iEE0NYHStSFn+n2YKNUXcK9VBSq4cF\ncXjvmiHELqMTOcLqogHwA62+4CH867iIbCE6XL5pwUugOZN/iHVdtu+AbF8NE9qorhAxdBJwXa7q\niFxVpRHzkY8QQXcyfpSpbFdyAKID6LWeDcHHbRxiMsgXzdfpuRCvDUvIiNGoe8KzMxqVLfQOarX6\ncVeuXIHtZMN5Q1hdkXeIn9C8FTKfLmN6r7nN+qXT8YZqCzGluY/WEpfRSdOBebJVmjOhbGzpO/FL\nczKv6f4/v2PcYrROz5eyAAAAAElFTkSuQmCC\n", "text/latex": [ "$$2.26492$$" ], @@ -486,9 +411,7 @@ { "cell_type": "code", "execution_count": 14, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { @@ -529,19 +452,11 @@ { "cell_type": "code", "execution_count": 15, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAABUAAAAMBAMAAAB/4Ov2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\n", - "ZkTTotXTAAAAdklEQVQIHWNgYBBUMmCAAtcE9oIbs87Pmn+BgaGRgTNhNsN9Bt4HDFy/GRi4FRjW\n", - "MLA5MHAcYGBg3sAQyMDHwLBfAKSP+x+I5AdqSWDgA6pkYOCawLDLgIExAMRmELyrwMDAMwHMBhMg\n", - "lTBgbwBjMbyMnwhkAwDTKhfdwCmMrwAAAABJRU5ErkJggg==\n" - ], + "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAMBAMAAAB/4Ov2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNgYBBSMmCAAtcE9oIbs87Pmn+B\ngaGRgTNhNsN9Bt4HDFy/GRi4FRjWMLA5MHAcYGBg3sAQyMDHwLBfAKSP+x+I5AdqSWDgA6pkYOCa\nwLDLgIExAMRmELyrwMDAMwHMBhMglTBgbwBjMbyMnwhkAwDTtxfeCPtWNAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$c_{T}$$" ], @@ -570,18 +485,11 @@ { "cell_type": "code", "execution_count": 16, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAELvv3c2rVESJdpkiZvtO\n", - "0HUAAABRSURBVAgdY2BUdmAAAtYEEMlXACL7BEDkFBDBELZqOwMD41MBvgMM7A8Z+CYwcAcwyE1g\n", - "YN3A0FfAwHeBYZ8AA18Doy0DA9sFpg1AlbtyGRgAGgIPXyFTKDkAAAAASUVORK5CYII=\n" - ], + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAELvv3c2rVESJdpkiZvtO\n0HUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABSSURBVAgdY2BUdmAAAtYEEMlXACL7BEDkFBDBELZq\nOwMD41MBuQMM7A8Z+CYwcAcwyE1gYN3A0FfAwHeBYZ8AA18Doy0DA9sFpg1Albv2MjAAAB2yD7/K\nbXugAAAAAElFTkSuQmCC\n", "text/latex": [ "$$b$$" ], @@ -610,9 +518,7 @@ { "cell_type": "code", "execution_count": 17, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "x, y, z, t = symbols('x y z t')" @@ -628,25 +534,13 @@ { "cell_type": "code", "execution_count": 18, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAaBAMAAAD8uecGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMkS7zRCZdiKJ71Rm\n", - "q90icBAQAAABaElEQVQoFX2QvUvDUBTFT0LS5tUo4j/Q4C4WRHCs4OAiFHQSoRkU10wVOkhBF11a\n", - "HBzqYDcHl04uLp0cFCdxctFBBxdRWz+wUs8L9L2UxlxI8rvnHO67LwAmkFCB9IxcQgJT0rQKSRHR\n", - "orv5T+Jsq+TRegScTnxkdxb2Ag8oAcKPjzzTXs0DVSDTjI2ID8qpT2CshWz8tlaXEfedsRwuiTk+\n", - "+x5fulQk3cA6wx6tYqB9kvjmSx5kdjBDkqYY3MnhGbB4W+MVS8AkcA+3IoO6jrljlZOdNzwBAQwf\n", - "Qrsh8SbYJjk/coqPdBN7gxHbzAPzdjiFkTlkfSwORgRu4fgcbXzhBqgfnv7Wa7DbkdTo1QHMo4vw\n", - "RmVguT1ePqH9wg37JXpduD1e2m1gWutmpR+IfPl3M57qXUURsAKkaqpfUxSBYgFGQ/UtRRG4Jt9F\n", - "+hh8oLYSo2vJlXuM+FoYpp1QOh82tLKhMYH+AIxDSfatV65XAAAAAElFTkSuQmCC\n" - ], + "image/png": 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"text/latex": [ "$$\\sin^{2}{\\left (y^{2} \\right )} + \\cos^{2}{\\left (y^{2} \\right )}$$" ], @@ -1079,19 +852,11 @@ { "cell_type": "code", "execution_count": 28, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAABYAAAAOBAMAAADZH/H+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\n", - "MqsI8sYEAAAAjElEQVQIHWNgEFIyYYACxgAG/wQom/0LA38DlM25kmH/ASgbSMHVANlHGRjq////\n", - "/5WBgVcjgIFJ112sUACkUuvCFQZF9gsgJgPXagYGA74NDAyMAgzMXxh4H3AABfl/MzB/Y2BN4AeK\n", - "cxkwcHxk2L+BfwIDA1sCg38BQwUDlwJQ0dRQSwYGMwY+BQYAbFgct5yPkD8AAAAASUVORK5CYII=\n" - ], + "image/png": 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"text/latex": [ "$$\\left(x^{3} + 3 y + 2\\right)^{2}$$" ], @@ -1358,34 +1035,11 @@ { "cell_type": "code", "execution_count": 33, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAScAAAAYBAMAAABKLya+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\n", - "75nQ6/gxAAAD1ElEQVRIDZWWT2gcVRzHv2//TLKbbjO00qOdphLwYBJtL566B6Ug1CxePGZdSkWQ\n", - "dAuCerAJtQgiwioUXYSSFvGgQhd6rJqcAl7MQltKi9BISqWsYrYisUVY37zf+838ZjdvRufw5r33\n", - "+X5/v+/Ovl0GoMtrNO1s9OZNPzu6mdhJU3i9XxLa/7G4UOg61b+i52QE0hQn8XyG24VzcypwMdzH\n", - "kpMRSFOcx0Y1w+7AlRUHoO0HqTSEbsVNzNcy7bsKJi/P+LsCs6l+djMiqYorKaXTCs9vFpedXD0d\n", - "OBmBdMW5DLcLH1zBX0PMa7wZ7TgO+p5Np+LJxj1mXqyyWyX96F7ufceC5D0uOlnDTpKpC+KcbKwk\n", - "oV2Nd6LtIYW6hRNVC89GIjsprPpQczhdGwZmHRctNflJqSpJS3Vco5k+L/NNmpqRFcAzNpRQPEHC\n", - "fBd5C4v14hptWqZ+OOYj76PSpW0zjhYFip1yiyTlgO5LVbrr8Tze86MFwAoU3rV9heIyCSdb8Po0\n", - 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"text/latex": [ "$$x^{6} + 6 x^{3} y + 4 x^{3} + 9 y^{2} + 12 y + 4$$" ], @@ -1407,33 +1061,11 @@ { "cell_type": "code", "execution_count": 34, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAANMAAAAzBAMAAAD/bSBoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjKJ\n", - "RFRer8KoAAADkUlEQVRYCe1WS2gTURS9M2ObfzTiZyGWVwS7cRHQbgQxqN3oIlHSuq0ouBC0VGhd\n", - "FJqFC0FKK36qIhoXghYXdeFOSfxUoRSanV9oUEHcxBgXRYvG95kJbyZ3JpOJ7apv8ebec899582b\n", - "m3cDUDcUUgctE6AlV0pK7TmzUlIAo6tSrRXM6gG2dn4uKzCcPtKiDk23/VZP5bVfwDHZbWD78Lid\n", - "VCgh8y9DLi77znakH43bSSlEpo/B3oLsN7BnsXjf+LUshsNRC/qdWAAnN9AMWasAaHMvi7UFJ2uW\n", - "MLrThyyI7ioEwJfCYygapuRuiP42gsyXh9oP8+iR8lahlWVuA9tfpMVZhF8G7Z5h6M92Av6MjG0T\n", - "jt4qbsqhBnaOALwtqD90WiQfmTZl+CegfUlGvhoOr7O04bl43uec2gE+Kn0hpiyl4ih1zkR2dg7z\n", - 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"text/latex": [ "$$\\frac{1}{\\left(x - 1\\right)^{2}} \\left(3 x^{2} - 2 x + 1\\right)$$" ], @@ -1459,30 +1091,11 @@ { "cell_type": "code", "execution_count": 35, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAL4AAAAzBAMAAADbdK+oAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\n", - "MqsI8sYEAAAC2ElEQVRYCbWWy2sTcRDHZzfp7uaxTbAg6ikUoiIiFR8gHtqD4kkakF5le1cIvQge\n", - "NAeh4sUiKF7Eq1WQFQ8FkboHPRlp/gM9iSi0VYpUAsbfZp+/zcwyP2n2kMzjO5+dndndBCD/sFv5\n", - "+VRWWzjVS7k805rh6YRqHfQ/bHEkvMnn/wB4HpVxv7VrfP4zgC3VAelVPn/DVecvKvDFTO673MEE\n", - "ukpbiV/6pYYHHZT4Cjdb0MhlNf4VxfYrDSW++UWRX9zc3HrZZhddBK3OFofCMv/+FK8SU5lf4/MP\n", - "TDe/qbavz/51uDUbg8Euot03fRqJ7llIa8G8s2e0UZC5A7VOEH41msUiTFlYWlqFd15gX8JoozGm\n", - "LCmM5sMsZMoS/ofQZBYyZRHfPtoaKx/gWC84AbMxpizqH6D8QuYPiCNzmYRKDvuvpMKOqDze7d7r\n", - 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"text/latex": [ "$$3 + \\frac{4}{x - 1} + \\frac{2}{\\left(x - 1\\right)^{2}}$$" ], @@ -1505,27 +1118,11 @@ { "cell_type": "code", "execution_count": 36, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAALYAAAAWBAMAAACfwnvZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\n", - "me8Q6PJIAAACVElEQVQ4EY2TP2gTURzHv3eX3CXpJT2iuNmmEbpapYOLeIvoVIqDLoIxhRTBYDfd\n", - "BDcXjV3qKdIq4iCIcRbxJnFLBrciKAjqItdijQThfJffu78vL+QN7/fe7/P9fe/37t0BNJprFl+J\n", - "of1OzkS1kClvlLtCkid0W+vI2DT5csPwZLqCbf6WsenyZWlvldXcwXQeMtWcKyNAaU/OpiDVzQmi\n", - "Qm0CnAJl7/Jo81NU1Y5WtMh9fuvi4+lWq5EBwXYMNIcpnfIdKzbPGP0UAg5BP8Bt3/eXMiDYCnDO\n", - "zQ1Id5iC9gVal5a4iMu0UmyKJ4BfYMfSuGAUpLBgG19J+IzCbAc6v0Hz0cs3lCzVKP4AzrgN4Dpt\n", - "aZZCs7lmk4R7z3eh8i+v4vteunzbYt5AvkZpOaweO7uYEHHvCut7gAwLW2Nm9y1gBlDqyxeaae8Y\n", - 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"text/latex": [ "$$x^{3} + 9 x^{2} + 27 x + 27$$" ], @@ -1548,23 +1145,11 @@ { "cell_type": "code", "execution_count": 37, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAbBAMAAAAzEOSeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMmYiu80QdonvRN2Z\n", - "VKvu110NAAABbElEQVQoFYWSv0rDUBTGv6SmrU2jHd2M+gJBXRSHog/QLtHF4YJD6SLd7GSDKHWQ\n", - "gCCCTsVNHHQTQbGDo0M2X6MI6iCiJ/efRiv3wM2533d+3HNyE8AQ3ZgZCJT9ctXIRPbAxADluplp\n", - "VoxMY82IUK8qYPn4J6a4b30A88OI3P5ugjGapFnJPQOrw5gleFSiN3ID+xiFKMNsCNVJ8A60aZJu\n", - "HGA0yDBPQp338i/USJYuMggkA97LSUTxgFJje7HdElIzTh8o0kpjGchH9rVL86WhmIlbErY07whn\n", - "xVcv4IhmUDykflfCPKVz4FSF+D4HOGGwBpoBxltc7IXhTRiu034OqCWaoV7AZAUszfqcT4aaj5Ls\n", - "RTN7rIMC44ia5wi4D/TMMXCZ1DArEMVMw3ujT94XLt3h5lZjx88ypceHHuDQSsMNRJZPdT9cLkjT\n", - "7suNSGc/VVeJFbX5k/N1ZQ39x3hxRI+R8xX9O89I4wvTnkk/yb5kGAAAAABJRU5ErkJggg==\n" - ], + "image/png": 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"text/latex": [ "$$2 \\sin{\\left (x \\right )} \\cos^{2}{\\left (y \\right )} - \\sin{\\left (x \\right )} + 2 \\sin{\\left (y \\right )} \\cos{\\left (x \\right )} \\cos{\\left (y \\right )}$$" ], @@ -1793,23 +1284,11 @@ { "cell_type": "code", "execution_count": 42, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAVBAMAAAATJmVgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnZUu0TviSKZ\n", - "ZquXRLrEAAABSklEQVQoFaWSPUvDYBSFTz6a1ITYUOziIBVEdHIQXc2i4NQidBH8AHHP0kkkgj/A\n", - "dHOSjA62vIOgi+jg6BD8A2bQubhL8b4mNwmazQvJeXI4OfcNBPjn6HFa4Hi/i7a7hXPH+M7A2omY\n", - "gHlGlaFCaxGbBbFT6JRgVkKmTG+93DjPCYsFYvN4xw4Sffm59yHdG7pa+/e79D0Ss1F8aDATqH08\n", - "ymWngBI5a1obeKXHbOoDWDATmCEaMXnrgOVanwa9cYTmqpwFsjf64ifVRoO24JK6oPpE6MhbNrMr\n", - "XzCpy89TSKmcsgXezkop2kgtHlyg9LlGhOm4lKLTG+4BapQqnd4YwBS0jjeOgIvDAA/U+CJr06kP\n", - 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"text/latex": [ "$$\\frac{\\partial^{3}}{\\partial x^{2}\\partial y} \\left(x^{2} \\cos{\\left (y \\right )} + y^{3} \\sin^{2}{\\left (x \\right )}\\right)$$" ], @@ -2130,22 +1492,11 @@ { "cell_type": "code", "execution_count": 49, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAACoAAAAVBAMAAADCwWEQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdrur781mRIlUMhAi\n", - "3ZlAc+EGAAABEklEQVQYGYWQQUsCURSFj+BMz2aG1DbuCgxXQUGtxMXs3AouIlw4m+hXhILQMl0X\n", - "gf+gEWk/uNCtLtvND4ioIEok8tz3fG67MPec882bdy8D/FO7kTngDkWLf+Wzow9gYiBQ1eaLvQK0\n", - "LHXE7PywXSPTlSClnVrS3SObCJHyQjZnBR8PuBFg6pKyN8eUcsdntD8r5Y3tpKiRcJ7XdRdBA7hi\n", - "7DfPQ8oLoGK1yiXAI2M79k4pdZ6Fc0iHJz6/8CIKKWfwVk31urS8gXmIGLKO+paop+XiA2RIOS14\n", - "N7QHdAYnesk3jNufrxrzs3FhdBsxXJhz0oPEej+0DnBT65VeZJOOLX22RnT717evhfqRdF6VYA0C\n", - "4jc5yL+/0gAAAABJRU5ErkJggg==\n" - ], + "image/png": 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"text/latex": [ "$$\\frac{d}{d x} G{\\left (x \\right )} \\left. \\frac{d}{d \\xi_{1}} F{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=G{\\left (x \\right )} }}$$" ], @@ -2277,21 +1585,11 @@ { "cell_type": "code", "execution_count": 52, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAACcAAAAUBAMAAAD8YzkFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\n", - "VGZoascqAAAA7UlEQVQYGWNgQAM1Kscc0IQYeAKyuRoYGJWMkWVYGPYB1YUxsP9AUf4RyEtlYJiJ\n", - "LMjygYWBYRYDQ74DQlSefQE7A8P+ABTBsBM1R8FK1geE6zAwTmVgrTkkANPD8Z2h4CYDzweGLgHG\n", - "ApgglwHjBW0GZgPGuQxBDjBBZQZWhjkM/AfY5p15ChPjSWBg4PjK4O/AD2TAwCUGRgGmDQwnGfjh\n", - "BjKwKDDwCDAnMJgxcAIFD0CUSpUXZTEwGXD8Y+DZwBAkABHc////LwbW9ENALx869wAiBiOBpqKD\n", - "bIb4BnQxhjUMNhhiDD2HHJAFATXPNMuriUczAAAAAElFTkSuQmCC\n" - ], + "image/png": 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"text/latex": [ "$$\\left(\\frac{x}{\\tan{\\left (x \\right )}}\\right)^{\\frac{1}{x^{2}}}$$" ], @@ -2662,19 +1825,11 @@ { "cell_type": "code", "execution_count": 59, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAACEAAAAWBAMAAAC8tehJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\n", - "zbsXyEShAAAAjElEQVQYGWNgQAEbUXhATlIrughDKa1EFn+ZgGE0TIDzHRA8ZOD/////B5gYhF5j\n", - "vQBVgOEZ5wU0EQEmBaAIo7JJAEyCSw/ESg9gVYCJMLABdbF2MogsgIkYMHxlYODuMT4IE2Bs4Gpg\n", - "YODfAOMD6ckg2/kVkETATA6giAGKIPMFBhG47RAZE5sDKEoA0TYeyV9YhCkAAAAASUVORK5CYII=\n" - ], + "image/png": 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╲╱ 3 ⋅⎜x - ─⎟ ⎜x - ─⎟ ╲╱ 3 ⋅⎜\n", - "x ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝\n", - "─ - ──────── - ────────────── + ──────── + ────────────── - ──────── - ───────\n", - "2 10080 1440 240 48 12 4\n", + "image/png": 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3lyDNO8mBjTeHC+QOdKH9LJ442ngHKh4XB7u86spFmaBgzU0mQdcgQUEKNgmbdEYenpKh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\ne0Er/EIAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$\\frac{x}{2} - \\frac{1}{10080} \\left(x - \\frac{2 \\pi}{3}\\right)^{7} - \\frac{\\sqrt{3}}{1440} \\left(x - \\frac{2 \\pi}{3}\\right)^{6} + \\frac{1}{240} \\left(x - \\frac{2 \\pi}{3}\\right)^{5} + \\frac{\\sqrt{3}}{48} \\left(x - \\frac{2 \\pi}{3}\\right)^{4} - \\frac{1}{12} \\left(x - \\frac{2 \\pi}{3}\\right)^{3} - \\frac{\\sqrt{3}}{4} \\left(x - \\frac{2 \\pi}{3}\\right)^{2} - \\frac{\\pi}{3} + \\frac{\\sqrt{3}}{2}$$" + ], + "text/plain": [ + " 7 6 5 4 3 \n", + " ⎛ 2⋅π⎞ ⎛ 2⋅π⎞ ⎛ 2⋅π⎞ ⎛ 2⋅π⎞ ⎛ 2⋅π⎞ \n", + " ⎜x - ───⎟ √3⋅⎜x - ───⎟ ⎜x - ───⎟ √3⋅⎜x - ───⎟ ⎜x - ───⎟ √3⋅\n", + "x ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ \n", + "─ - ────────── - ───────────── + ────────── + ───────────── - ────────── - ───\n", + "2 10080 1440 240 48 12 \n", "\n", - " 2 \n", - " π⎞ \n", - "x - ─⎟ ___\n", - " 3⎠ π ╲╱ 3 \n", - "─────── - ─ + ─────\n", - " 6 2 " + " 2 \n", + "⎛ 2⋅π⎞ \n", + "⎜x - ───⎟ \n", + "⎝ 3 ⎠ π √3\n", + "────────── - ─ + ──\n", + " 4 3 2 " ] }, "execution_count": 64, @@ -3102,22 +2045,11 @@ { "cell_type": "code", "execution_count": 65, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAUBAMAAABvx07iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\n", - "me8Q6PJIAAABM0lEQVQoFWNggIDcvQJQFpEUmwPzBCKVQpVxOPB8IU0HXwDLV9J0MDBwfSRVB4cC\n", - "qTpyidDAkpa9AK6M/QKciZvhy8D1HS4bxhADZ+Nk6C9g+AGT5Jm1aiOMjZuOC2D8CZPl+///A4yN\n", - "jwa7SkjZVU0BUxWjkklQEoYwJ9C7jAnsTRxYUogYQ4VCJ0hH5kwQmAvWfKwXSLEJsH3kcgDzUYhE\n", - "hnyBJSgiIA7bG6AdDJwb4BKMMzqAoPMAUECAYTZcGInxTgDI4VdAEkFifkJiQ5guDAz7FwCZ5w8A\n", - "DcQEPH+A9gNBBsjajmYQ878Aw/4CBi4BfQYmLDoWMn1gUAQpQwKvGRj6HRjiF+xncEcShTJZf3F/\n", - "YClAEy9n4AI6VVhJKB1dBqiQ0UhRrRxNAwPr3dsBDAAbFkHpklxB3gAAAABJRU5ErkJggg==\n" - ], + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAUBAMAAABvx07iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABMUlEQVQoFWNggIDcvQJQFpEUmwPzBCKVQpVx\nOPB8IVFHAMtX0nQwMHB9JFUHhwKpOnKJ0MCSlr0Aroz9ApyJm+HLwPUdLhvGEANn42ToL2D4AZPk\nmbVqI4yNm44LYPwJk+X7//8DjI2PBrtKSNlVTQFTFaOSSVAShrBEAgMDYwJ7EweWFCLGUKHQCdKR\nORME5oI1u/UKMDCwCbB95HIA81GIRIZ8gSUoIiAO21OgHQycG+ASjDM6gKDzAFBAgGE2XBiJ8Q5o\nCQO/ApIIEvMTEhvCdGFg2L8AyDx/AMJHI3n+QAQyQNZ2NIM4/wVAOrgE9BmYQFahgYVMHxgU0cRe\nMzD0OzDEL9jP4I4mA+Sy/uL+wFKAJl7OwAV0qrCS0HV0GaBCRiNF3XI0DQysd28HMAAA7/dBoTLR\np+8AAAAASUVORK5CYII=\n", "text/latex": [ "$$x^{2} - x = 3$$" ], @@ -3140,23 +2072,11 @@ { "cell_type": "code", "execution_count": 66, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAIMAAAAUBAMAAABc0JOuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - 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"text/latex": [ - "$$\\begin{bmatrix}\\frac{1}{2} + \\frac{\\sqrt{13}}{2}, & - \\frac{\\sqrt{13}}{2} + \\frac{1}{2}\\end{bmatrix}$$" + "$$\\left [ \\frac{1}{2} + \\frac{\\sqrt{13}}{2}, \\quad - \\frac{\\sqrt{13}}{2} + \\frac{1}{2}\\right ]$$" ], "text/plain": [ - "⎡ ____ ____ ⎤\n", - "⎢1 ╲╱ 13 ╲╱ 13 1⎥\n", - "⎢─ + ──────, - ────── + ─⎥\n", - "⎣2 2 2 2⎦" + "⎡1 √13 √13 1⎤\n", + "⎢─ + ───, - ─── + ─⎥\n", + "⎣2 2 2 2⎦" ] }, "execution_count": 67, @@ -3232,22 +2133,11 @@ { "cell_type": "code", "execution_count": 68, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAE8AAAATBAMAAAAjVk05AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\n", - "VM0GsGrEAAABKUlEQVQoFWNgQAIiJpuQeHiYF7qY8cgiS7kgc/CwDVbw4pFFktJ2dkXikcxk91LS\n", - "ZjRA1sb/HwiQBSBszw0MvCc3YIqji1SlA0XOC6AKMyqg8kG89wVAohNNnNsBTYCBgecvSCgGiBmV\n", - "XYGGF7scBLLNToNEUYB8AohbCsSdAlwfGDgSGYSBbDaQIBDYvQOBNyCmvQOIBAL22Qw8CxgkL7Af\n", - "BXLqwEIoRH4AlMvznYHNgUFzizfIzXe4URSBOPPBIkBJzgUM9QEMnyAKhBwhNBKZD2LzAo2tN2C4\n", - "z1DwGUkKyDRLA4EkkCA4BHmAjPoLDOt4BSYzMLCAhDGBvAFQzByI+Ry4k5gZ/BjYQcGDBfD+Bjod\n", - "5H52FVMhDwaWI7uxKAIL1SzSaQAyAPt1PB+SzWHeAAAAAElFTkSuQmCC\n" - ], + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAATBAMAAAAjVk05AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLUlEQVQoFWNgQAIiJvOQeHiYF6qY8cgiS7kg\nc/CwDZZz4pFFknrp7IPEI5nJ5vXuJaMBsjb+/0CALABhu05g4FWdgCmOLpK1DSiiL4AqzPgAlQ/i\n9ScAiVIQCwlwOiBxIEymvyD6EBAzKquEMTAkuygC2daaIFEUIL8BxE0E4lIBjg8M7BsZhIFsbpAg\nEFh0gEATiGnvACKBgC2SgWkBg+gFNh0gJwMshELsPwDlMn1j4HZgeDrFMwEocAkzZuJB4gxAgnUB\nQ/4Bhn9ADhDIOUJoJHI/iM17gIEh34DhPgPDZyQpINN6NwjsAwmCQ5AJyMi/wLCCVyCYgYEFJIwJ\n5EGRZwbEHA6c+5gZPBjYQMGDBfD+BDo9ASjB9sRGzp2BRWUmFkVgoaR1jwqADADyDDxaVpyiwAAA\nAABJRU5ErkJggg==\n", "text/latex": [ "$$a e^{\\frac{x}{t}} = C$$" ], @@ -3273,29 +2163,13 @@ { "cell_type": "code", "execution_count": 69, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAbBAMAAAAKd1XFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iEFSZzYlEq3Yy\n", - "72ZZXIwmAAACH0lEQVQ4EX2UP2gTYRjGn+Qul15yiVFwCQghOhQtJR10cOlRRASHxMVFoTcIFgfp\n", - "4KCLZooVHTo7aFUQpA5xkUKWuFYlXQQFhwpOulj8g4VIfJ7vLtc0IXnh3vd5n/vd+335kgswOfLF\n", - "pfOVGDlU3tOxaYS3Cli3YR8BmjTc8jRK+4mB7g11hw98AR4a+/QE+AaJj7zeAlPboifADtfO/iJz\n", - "htex/fDmTfUDMe8DyUJkvFYNJy9UAPt45PdLg+JAEHXzqiG8KFlUGogP1Mk6D2RbQimEr1INw562\n", - "63Abjh+JCJ5mOwznt2TOwn6gmlWnye6TnXVKbuNc6x7PtPH9FVtnmQlTz+/7qq7WMdtI/VBfhHcU\n", - "iSbt2SzbREFmP7wulYHDIUV9HK+bKeGpkPSacj+sXSoD6zNrcqcG/HW2QjgZ0OuZ4MLWb3YGrq5S\n", - "EV5k3bG6tg5t7OTHsA2syX+sjY32KOz9pGcm38VlA+cCuLspswyQKeiRfuydxjW7SfMgvM9I191b\n", - 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"text/latex": [ - "$$\\begin{bmatrix}t \\log{\\left (\\frac{C}{a} \\right )}\\end{bmatrix}$$" + "$$\\left [ t \\log{\\left (\\frac{C}{a} \\right )}\\right ]$$" ], "text/plain": [ "⎡ ⎛C⎞⎤\n", @@ -3339,39 +2213,11 @@ { "cell_type": "code", "execution_count": 70, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAT0AAAAvBAMAAABj3ACKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZdiLvVM27RDKr\n", - "Zt3KPpNmAAAEy0lEQVRYCb2YXYgbVRTH/zO7+d4kw1qKDwuNbkSQFoNWWPGh03QRH5TOU0VR2EVL\n", - "VRAi+PGmUYSuPmjE9UGoNEhBRHDzoH0QdBepIPgSEEEQ21Aq+lRTP1hQZD33ztzJfCW5Z7PZ+zD3\n", - "zL2/859/7syQOweYVlt815qW9F7oms5ccy90pqVRcgp/T0t7L3TT7dQ/e6EzPY3Mjelpx5U/fiI+\n", - "NnxE0CV7+PwUZrZZmkQvshImhfOsu0V0sTbpJVn5Zo+DE/0hPuFkTMpmWcuRrRXOvvTOpNfUz3+s\n", - "utng0emdnb5+xoRk/jTOW9oaPFpbdgR4sIulEdORKR4dSd7V6TELV/UTebS+7nDyaRiMPysePfyq\n", - 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"text/latex": [ "$$y{\\left (x \\right )} = \\frac{1}{\\sqrt{e^{x}}} \\left(C_{1} \\sin{\\left (\\frac{\\sqrt{3} x}{2} \\right )} + C_{2} \\cos{\\left (\\frac{\\sqrt{3} x}{2} \\right )}\\right) + \\sin{\\left (x \\right )}$$" ], "text/plain": [ - " ⎛ ___ ⎞ ⎛ ___ ⎞ \n", - " ⎜╲╱ 3 ⋅x⎟ ⎜╲╱ 3 ⋅x⎟ \n", - " C₁⋅sin⎜───────⎟ + C₂⋅cos⎜───────⎟ \n", - " ⎝ 2 ⎠ ⎝ 2 ⎠ \n", - "y(x) = ───────────────────────────────── + sin(x)\n", - " ____ \n", - " ╱ x \n", - " ╲╱ ℯ " + " ⎛√3⋅x⎞ ⎛√3⋅x⎞ \n", + " C₁⋅sin⎜────⎟ + C₂⋅cos⎜────⎟ \n", + " ⎝ 2 ⎠ ⎝ 2 ⎠ \n", + "y(x) = ─────────────────────────── + sin(x)\n", + " ____ \n", + " ╱ x \n", + " ╲╱ ℯ " ] }, "execution_count": 71, @@ -3497,23 +2282,11 @@ { "cell_type": "code", "execution_count": 72, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAADcAAAAyBAMAAAAKOF7GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhAiq3bdRLtm\n", - 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"text/latex": [ "$$\\left[\\begin{matrix}\\frac{1}{a} + \\frac{b c}{a^{2} \\left(d - \\frac{b c}{a}\\right)} & - \\frac{b}{a \\left(d - \\frac{b c}{a}\\right)}\\\\- \\frac{c}{a \\left(d - \\frac{b c}{a}\\right)} & \\frac{1}{d - \\frac{b c}{a}}\\end{matrix}\\right]$$" ], @@ -3675,34 +2384,11 @@ { "cell_type": "code", "execution_count": 75, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAKMAAAAzBAMAAAAJJKXwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhAiq3bdRLtm\n", - "mc0lg57xAAADxklEQVRYCe2XTUhUURTHzzx9iqOOUhGUlUNEGGQMSBG0UMiCNjkuqkXIzE5aJbVx\n", - "ETGEUARBH/SxtGhXkC1aWrMsKHBRtBBsIMJa+BBNFyJM59xzzn3vvhlDHXd1F+/de/7n/N6d+4Y5\n", - "/wGwo3sya+c1TTrLC1zv5RtLPKvh6pXL7bC9/xQjmvL1yzXAuNTvP9kOOxSTGG9Y0XkN9/oIEqBu\n", - "yUVd3OWu7apCkEDy9RS4yKaCLeLJbGxtlxWCBNoyMWS3lPh5mSxaRmxihfNu5qusi2yZEr2uwJNk\n", - 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"text/latex": [ "$$\\left[\\begin{matrix}a^{2} + b c & a b + b d\\\\a c + c d & b c + d^{2}\\end{matrix}\\right]$$" ], @@ -3787,10 +2473,8 @@ }, { "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false - }, + "execution_count": 76, + "metadata": {}, "outputs": [ { "data": { @@ -3799,7 +2483,7 @@ "" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3814,14 +2498,16 @@ }, { "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false - }, + "execution_count": 77, + "metadata": {}, "outputs": [ { "data": { "text/html": [ + "/* This template is inspired in the one used by Lorena Barba\n", + "in the numerical-mooc repository: https://github.com/numerical-mooc/numerical-mooc\n", + "We thank her work and hope you also enjoy the look of the notobooks with this style */\n", + "\n", "\n", "\n", "El estilo se ha aplicado =)\n", @@ -3899,7 +2585,7 @@ " font-weight: 600;\n", " font-size: 35pt;\n", " line-height: 100%;\n", - " color: #234765;\n", + " color: #000000;\n", " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " display: block;\n", @@ -3912,7 +2598,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #234765;\n", + " color: #2c6391;\n", "}\t\n", "\n", ".text_cell_render h3 {\n", @@ -3922,7 +2608,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #376f9e;\n", + " color: #387eb8;\n", "}\n", "\n", ".text_cell_render h4 { /*Use this for captions*/\n", @@ -3932,7 +2618,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #376f9e;\n", + " color: #5797cc;\n", "}\n", "\n", ".text_cell_render h5 { /*Use this for small titles*/\n", @@ -3981,10 +2667,10 @@ "\n" ], "text/plain": [ - "" + "" ] }, - "execution_count": 1, + "execution_count": 77, "metadata": {}, "output_type": "execute_result" } @@ -4013,9 +2699,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.3" + "version": "3.5.4" } }, "nbformat": 4, - "nbformat_minor": 0 + "nbformat_minor": 1 } diff --git a/notebooks_completos/040-SymPy-Mecanica.ipynb b/notebooks_completos/040-SymPy-Mecanica.ipynb deleted file mode 100644 index 146faaa..0000000 --- a/notebooks_completos/040-SymPy-Mecanica.ipynb +++ /dev/null @@ -1,2776 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\"AeroPython\"" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Clase 5b: Mecánica con SymPy" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "_Si SymPy te ha parecido hasta ahora un CAS decente e incluso interesante (nada como tener los resultados en $\\LaTeX$ incrustados en el notebook y la sintaxis de Python para hacer cálculo simbólico) entonces espera a ver el paquete `mechanics`. Con él, podremos manipular velocidades y aceleraciones de sólidos expresadas en distintos sistemas de referencia con una facilidad impresionante._\n", - "\n", - "_Tienes disponible la documentación de `mechanics` en http://docs.sympy.org/0.7.5/modules/physics/mechanics/index.html._" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Sistemas de referencia" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "El objeto primordial que vamos a manejar van a ser los sistemas de referencia. Podremos definir relaciones geométricas entre ellos y de esta forma las transformaciones de vectores entre un sistema y otro serán triviales." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "IPython console for SymPy 0.7.5 (Python 3.4.2-64-bit) (ground types: python)\n", - "\n", - "These commands were executed:\n", - ">>> from __future__ import division\n", - ">>> from sympy import *\n", - ">>> x, y, z, t = symbols('x y z t')\n", - ">>> k, m, n = symbols('k m n', integer=True)\n", - ">>> f, g, h = symbols('f g h', cls=Function)\n", - "\n", - "Documentation can be found at http://www.sympy.org\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "WARNING: Hook shutdown_hook is deprecated. Use the atexit module instead.\n" - ] - } - ], - "source": [ - "from sympy import init_session\n", - "init_session(use_latex=True)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Todo lo que necesitamos está en `sympy.physics.mechanics`, incluyendo la clase `ReferenceFrame`. Nada más crear un sistema de referencia podemos acceder a sus versores unitarios: `x`, `y` y `z`.\n", - "\n", - "http://docs.sympy.org/0.7.5/modules/physics/vector/vectors.html" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "from sympy.physics.mechanics import ReferenceFrame" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAABYAAAAUBAMAAAB7UtADAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAECLdzUTvdomZZjJU\n", - "u6tkz/T2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiklEQVQIHWNgYGBkQABlRzibNcVMAcaRL+BI\n", - "gLGBygVgbGS62cVSAcrn+sawPwnK5vnGwPsTpm4jA+cXGFum9zGMzfO/ES6u/wuk5vB/gXxDhvhP\n", - "DLxANfeDUhkY5L8zbPmswMD87QADA9Ozew33vzMwfUM4hPe9Icxghnesf2HsZd8Z/ufAOCAaAJAn\n", - "JoxLDW8EAAAAAElFTkSuQmCC\n" - ], - "text/latex": [ - "$$\\mathbf{\\hat{a}_x}$$" - ], - "text/plain": [ - "\u001b[94m\u001b[1ma_x\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 3, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A = ReferenceFrame(\"A\")\n", - "A.x" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Y para definir vectores solo tenemos que **multiplicar cada componente por su versor**:" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAWBAMAAABzm9yKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\n", - "VGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOElEQVQoFW3RMUvDQBQH8P+RS1tDjXURdWnU\n", - "D2BB6Jp8A29ycemigovgIm5dnaxIBwWprmZIB6cuyeQsLo71G4igNKnCGZNQ8857y9378X93RwLk\n", - "JYpVXVRnvir5RMnZ+pYHnI0u1LOyvuQ7qMYwv3GqC5Z9D7gCBExdsOzXwJGnDSkYijRY9Udrimet\n", - "4oF46/GkoUlSr00w7sF1NEHqVgtcYHOoCVLfAMyXdvAX3L3/rYd0kHi9A4wnQnsi8WewRfmaXj0f\n", - "fM5NyZ9kmduDPk+64A7qy7IBdwgjXumSh9Zyb96iD6yeHB+aMuLhHRDckBwKr8S2A4RSTrF0eWDI\n", - "CM10kFTu/MtSPnKbPnE24+7Pttmm0jlvUSm67XfC9iBaSJ4IFY2hn1ejLLI81bR99eNR6/+Q+U5q\n", - "P3bPZGZrDwqmAAAAAElFTkSuQmCC\n" - ], - "text/latex": [ - "$$2\\mathbf{\\hat{a}_x} - \\mathbf{\\hat{a}_y}$$" - ], - "text/plain": [ - "2 \u001b[94m\u001b[1ma_x\u001b[0;0m\u001b[0;0m - \u001b[94m\u001b[1ma_y\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 4, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "2 * A.x - 1 * A.y" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "De ahora en adelante, para trabajar como si nos enfrentáramos a un problema de la escuela, vamos a hacer dos cosas:\n", - "\n", - "* Definir un sistema inercial $1$ del que partir, para así poder referir todos los demás sistemas a él.\n", - "* Que los versores de ese sistema sean $i, j, k$." - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAE4AAAATBAMAAADMlCYHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu+9UzSKrMpkQdond\n", - "RGZc8tcOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVQoFWNgEDJgQAFFJ1G47P9/gfnM/x1Q\n", - "xBnyIeJwwSIIn/V/AlwIzFj/F5XPAtVXDBWeCqWZL0AYnAEQGqYOKs1gBWNAaeYFEAZUHT/MXqg6\n", - "3v+/IfJI6uyBalj9EyDiMPNiMNVNygQq6UdTx4ihjlUbZBJhdfcfEKXuC8QHYPO2uLgccnHxAOkD\n", - "28vt4uKj4uLSAOSz/Hn/D0gRtvcz139QUBJ032cG+QPEqYv/wcDA//8/UDEQwMIPLVzY//932P//\n", - "K0QNQh1nAZo6hAIo6x6Y5v/O9BkiwPsAKoGV4v9bfwCrBJogr8+JCWhCYC4ASv1WiKruNuQAAAAA\n", - "SUVORK5CYII=\n" - ], - "text/latex": [ - "$$\\mathbf{i} + \\mathbf{j} + \\mathbf{k}$$" - ], - "text/plain": [ - "\u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 5, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A = ReferenceFrame(\"1\", latexs=['\\mathbf{i}', '\\mathbf{j}', '\\mathbf{k}'])\n", - "A.x + A.y + A.z" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Y para no tener que hacerlo siempre, un pequeño truco de magia:" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "# Definimos nuestra propia clase para que los versores sean IJK\n", - "class IJKReferenceFrame(ReferenceFrame):\n", - " def __init__(self, name):\n", - " super().__init__(name, latexs=['\\mathbf{%s}_{%s}' % (idx, name) for idx in (\"i\", \"j\", \"k\")])\n", - " self.i = self.x\n", - " self.j = self.y\n", - " self.k = self.z" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAGcAAAATBAMAAAB/3A4dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu+9UzSKrMpnddhCJ\n", - "RGZaefImAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLklEQVQoFWNgEDJgwADsmhswxf7/gosx/3eA\n", - "s+EMtv8L4Gw4owihifV/AlwYzuD6fwHOhjNYEJoYiuGiQMZWKMcWiyCyJmRpBisUHpQDFUTSxI/i\n", - "PKj8/f8HkHQjNNlDFbP6J2DKs87HrmlTNlRtP1BTIUwfzHnyBxjYBdAEWX6x6gCFwOJATRWZaPIM\n", - "8gfY2yagCbL8uv8AqAcsDrLJEk0eqImBGUPTF7CPwOIITUdcXCa5uHiADEBoQgiy/Hz/DyiFpgko\n", - "guQnTJs+c/0PIF0TgzzQyaTaxBD/A6KJ//9/4gKC/f9/h/P/vyICCC30tiMFBJJHgUwggPvVCMJn\n", - "YLgHZvwv0DdgYFZAFYTxYOIR87NgQmD6//WfDKyd3xagCEI5uMQZrszsxqYeLgYAKb1mXNaAdd8A\n", - "AAAASUVORK5CYII=\n" - ], - "text/latex": [ - "$$\\mathbf{i}_{1} + \\mathbf{j}_{1} + \\mathbf{k}_{1}$$" - ], - "text/plain": [ - "\u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m + \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A = IJKReferenceFrame(\"1\")\n", - "A.i + A.j + A.k" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Álgebra vectorial" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Nuestros vectores funcionan también con símbolos, y podemos realizar las operaciones de producto escalar y producto vectorial con ellos." - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "R, V = symbols('R, V', positive=True)\n", - "r1 = R * (A.x + A.y + A.z)\n", - "v1 = V * (A.x - 2 * A.z)" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAJcAAAATBAMAAABvkIKEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqu7IjJ23e9UmRCJ\n", - "zUQBtxRlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACAUlEQVQ4EYWSv2/TUBDHvyl+qVxqOerUrSZI\n", - "ZWBoWFC3ZGZJJBbKEktIXcsEbOmAxIJEJspEIhg6VBUZWqlj/gHUzkhRYGBEBbd11SL0ONvv3POP\n", - "gpe7+37O3/fevQfwt9rnrBjVq1FRNIrSoWALul6/B1sfCY3TykpQv13DjPZZEdHAR9IMJ0C7YWlP\n", - "9KWp24B1gqr+mioiSaAjzRQVM2dYE13Y46LrAysdvOc6ijmYMaueAbOnsht4y+VGB+i1uEpiDmbM\n", - "bmwCg6GbOWbavwrYlzjWY+GXg2Q2SH92PdjPYE29sv4A2PJh9UrNEkhmB3f53+7yk1+Uf/KAh6zx\n", - "4ipYb9ZIbI6hGtdAJ7ReE0r4xgiLrdjs8Z18f/U3Ki9JbI7Vh6fXQCc8pvUMp7G4w2RnW/n+aJ6X\n", - "JNLO7IKZgc6feGIJp5O3PWn2eTJ5MZl8iYxpnhZxYVaEzvlSNKbYTFE2GEozIjyzbi26TGlWhM5F\n", - "VbeMWfTMDv2b8QUUjknPbD7ESOxMmBnoXKBJI4h35vwElvy1crMpUAlVv9zMQDJr0+4js7lpUMPc\n", - "fd/Vmp4UrRt/5pi7+hbwcb38mAYqrY8O9am4IOORNzPyfvnODDUhPqaU3nHxjZM46s6bPuxN1rKQ\n", - "1SueKA96y4wyUX8/h7Ud+BkxV/yPp+0/nu+k+b+Sv/57vXVbEKm8AAAAAElFTkSuQmCC\n" - ], - "text/latex": [ - "$$R\\mathbf{i}_{1} + R\\mathbf{j}_{1} + R\\mathbf{k}_{1}$$" - ], - "text/plain": [ - "R \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + R \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m + R \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "r1" - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAGgAAAASBAMAAABFi4Y1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NVDLvmRCrIolE\n", - "ZrtDPvbtAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABb0lEQVQoFXWQsUvDQBTGv5aWxlTbIrgJis5S\n", - "EfcLDroWEdwsOAhOZnPTrE5WJ7cGB13dHBz0P9DFTbCTo4MFwVp9vrvkpWeSHtzdd+9+3+V7AWS8\n", - "eaLG7G1qyM00rQKbn36JlqU02t3D9xaEKB8kJqfPTKkDlzojWNQJqkMIAZWYsMfEBM8tIa39Avpa\n", - "CMukGJqxQFvuAt0WhLBM3RDuDVDPi6cCbYoJjleir+jN+wdUWLkqpyeuNwPEBJuc/SAyvTbwrNUx\n", - "m3aikrVOfgNCqMYVRzLQo6/TGdP2uYVHstgDhFCzP1w00GlP/7voS2taOCsLPBZbWuOIpxBKmY40\n", - "VJ9/0bcmnjGZU7wUOHNCqCV64qOGigPfELqnjGkDbpgQypsbxKbabzDWVPZRCCGE8mofsakyNJ7c\n", - "eJd3t/xvhFCeS5xKx5niWDzqRNl4iojfjok29atEYaaHTE/mvdSShtZT97nHFHTdPMvF/hUZ+gMg\n", - "y2wqMF1p8AAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$V\\mathbf{i}_{1} - 2 V\\mathbf{k}_{1}$$" - ], - "text/plain": [ - "V \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m - 2⋅V \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "v1" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "from sympy.physics.mechanics import dot, cross" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAOBAMAAABN+VgMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\n", - "iUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyUlEQVQYGVWPMQ4BURCGP7FrV5AVF9DoFCQO\n", - "QOIAtlCzDUpqDUfYhEZFKSqJC2w0WjfgBkKxrAgGkbz3F/Mm35/3zwz8ZfbDbs9l8ipC/R78sbzO\n", - "APtK7CJtylM4DR/6NVbCEipnXoPqmJLAqWYMJeIhn8bYe80IYSNphyNpjRvhbOQKafs0NSN5xlwK\n", - "OQW/pFzlowJYC5AR5Af6TjgetozBKbekKmq436WI3wKFSitnZCKyWNesbshhZmSsST91vn11YDcT\n", - "1/sabx/pL43+CN+kAAAAAElFTkSuQmCC\n" - ], - "text/latex": [ - "$$- R V$$" - ], - "text/plain": [ - "-R⋅V" - ] - }, - "execution_count": 12, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "r1.dot(v1)\n", - "dot(r1, v1)\n", - "r1 & v1" - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAO0AAAATBAMAAACO+5fVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92\n", - "VGZ6zyUAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADKElEQVRIDY2VT2gTQRTGv6S7aZNmk+BJ8NCS\n", - "epVUBE9CAoIipSRQLHjqotB6s548VOyKB3so1pPSU3P30OJfxMuCgpdCAgURPDQHD6JgtWmbWFPH\n", - "b2bzZ7O7aftgZ9/M+818mTdvNsChNlc+NOwKjp11dXq4mmj4Ivr0nRyQWKovPKzgvTgDXPxrGmLY\n", - "B0Jffspf4yGx5F+SU7+JhYV7iGe3y9CyrzHmh04iuk8wOQu9htA2XcOCLiw6HpuE8Y9D3SRKcrbf\n", - "asDULIaGGfkMhP2694F3DOZtYCmn3AF2x/m0LFRwvFIZVXoe0ii2OPdbo9DADgbXOGgF6r5Qeijl\n", - "gGwBGXIf+LjNsJ3es6JKxyFkZ1Z8B+jbRfgXEM8F6mZScp+YY373+BMK0B8zl+48t3RJyDx3kzHx\n", - "p6PW8fq50U0TA7vANY4yz5vuJR0wmwLqwIYNrBQR4aCesZyQbDu6YZPdbhITgbpJC8YFqNKRc6h7\n", - "igfabTHO1Orr8xUOb9n4KqO3LOCydGht3Svn2fOS/ZzdIhWumvziR5mbRAOJHN/hhv6WL22WTdv6\n", - "2Iv/RuIVR1ZNmWale+lBk2jrIvrST1K3TTYn8FVK4UcBiO2pNFN3tULZm6PAiXPSTpOZ4SPPg8eL\n", - "oVlZzc5+N5Tn2i8rP+Uj5X4dsrOiKoKkCYRqspq53211ugZ12xa36PI8dB4ckqPfVUDmWa0WTadH\n", - "nqfTJkevAvNlH9nRVRObDdeasuhXE0U5Eq5uybR36X6CXkC+osqZtW9KTp2vd78iBdaAlwzU1ajC\n", - "cubVnJSrIXwQF0y7W1czES/I6xttgFmspRTY3i97hq2G8EbdcC8ZqCuv74odYoJuq7nhA8wzx27d\n", - "ieUnLCB+LxINrYxI86sXpPsTBr9CXjJQV34vtuxx5q3Y0p1i+bh1M0JUcV08Yn2tU9xSXFKIVrV0\n", - "9hubnin6SATpRjL1CiJ3bf4/qOU0IYZXxG6XrgoENt7zDYJCuSDdIFKOuffbiwG+NEOxSm8muTN4\n", - "0CF7c07EWDuKYHwyu3g0ldzPjx6P5Fr6jbp99JLHImIj550LcCya0H/9FiHnyd/STwAAAABJRU5E\n", - "rkJggg==\n" - ], - "text/latex": [ - "$$- 2 R V\\mathbf{i}_{1} + 3 R V\\mathbf{j}_{1} - R V\\mathbf{k}_{1}$$" - ], - "text/plain": [ - "- 2⋅R⋅V \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + 3⋅R⋅V \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m - R⋅V \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 13, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "r1.cross(v1)\n", - "cross(r1, v1)\n", - "r1 ^ v1" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Podemos hallar también la norma de los vectores con su método `magnitude` e incluso normalizarlos con `normalize`:" - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAVBAMAAADiLT7tAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\n", - "uzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABdUlEQVQoFW2RvUsDQRDF3yXkcpczH41gIxzp\n", - "BDGBlCpJYS9W2hkQsRLS2si1FmJKO9NZWHiCNhZ+IOks8gcISWsVktOgYIhvJjFF1oGdfffmt7uz\n", - "e8A4rMJoGhPLmOIVwzKMecMxjQPTmnWc+qxjfts105t1TsTYkZSpAl45Kq2HWBptA+8/NDSumBe/\n", - "RLotpmwL1hDOJ2WyKC5D2jndU2ZZmK4PlDt4pYxzaMRCTjFhrDthCh3gIoddyhUObHC4IpRJJYR5\n", - "4xkDbpWD1ZBKkynPMWZulImAIx+4rsCWykKhAtRFyT6OL0wmyrdDOo8+HqSCsxa8hghhUhCGynum\n", - "s1nVowC7Bzf4Y46VST8BbIfLJ7eyBrgXRFY7DWWyRVhsCdn6rVbYfXCuikyyVCq/+OiGei24veqE\n", - "6V6OlfQDzLEfPk+qjwDpYSAWI9EMdB4z/A3ydl4/U4P9oQUmeS2Gvfa9you1o3B/RHHIN/OKWpG0\n", - "NVX/i18UPWk6HD1fSgAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$\\sqrt{14} R V$$" - ], - "text/plain": [ - " ____ \n", - "╲╱ 14 ⋅R⋅V" - ] - }, - "execution_count": 14, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(r1 ^ v1).magnitude()" - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAQIAAAAwBAMAAAAIpGlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiJ2q0SZVIlm\n", - "77uIHctDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAENklEQVRYCcWXTWgVVxTHz0tmonkvyQt+ZFMw\n", - "Mbrpon1WF26qeVu7qG8liNYMXbmopCA0XVQUKYWC0FdchOrCgOLHouShy6A8Cl25sXQR6cK8pS2U\n", - "JA0Sv2/PuTP3zp2ZO2fuoJALeXPuPb/zv3/mzryXA5Ae00KPdMo2P6ZpsWrLG2uuyt8aNQ5hCdwR\n", - "9Sccto2RErgrOjgfyztEJXBXdLvDtgZSAndFr6D8btrCaxsbJcMTi/+oBcL7Rml2RC3lXV2Usbay\n", - "D+DUGon09+jTNvw2LHTDBOH+inRw2IYaay7KhNe6sGtOOjjTM8oT4cBLqOPONBCv/H2eHAw8kQv5\n", - "Hy7KVE2HNUAO/N96+GkdWx/D+ESYkWc7Qw5Ofmxl40UXZaK/wj/poNbXo3nOUKdAOEgHnSIHDspb\n", - "L6Aa3V3p4HfWwYcAGpcOhlqMA40WKJ+ZwM2DyEElYBx499oAGpcOxoBxoFFykK/sjz3HFwANyHtQ\n", - "A8YBwP1mjEsHHcZBjJIDRrnyFuChcnCDdzD8B2icHFSbjIMYJQec8rUWfBI5qHQYB34LRl4CKJwc\n", - "fACcA42iA1Z5uj3UiRxUZ2fPLwU0sYz6Goy8BlA4Obg1O/vmGwsaLSkUHbDKW9b66c0O3wUY7kXl\n", - "mQtm+p8BKFw+BwDcN5JC6RSAUfZe3ZG7hVy9JyeWj6EuLAT4wxHhkQN8inKHQouU4SP5kzR4bv07\n", - "fGRXNroA/87bVH/85SkthzjMLR/sAjwQWOT9KW9itiajfFQEWQpgAV/zxKiKicQ8OcngsMWua1H2\n", - "loOkWDjra6VWffF1asWcZnAYEBdMII6zaCOIs1z0mZncaU7s8W37smXV1UGi9GZi9o4TRwf1xCkU\n", - "O/hJdFyNNYKq2CiG/Yb5HBQ78KdKOKgsjRY7gGvo4HPFFTuAlQ54PcWz10bwBbnV4nkwOjhyViWd\n", - "HHhz+xTPXhunX2OexLcdoLEXIO6+dLQq78F1paQc6HwqWAW6B1XpoLKHZPc3qTaF0XQVGg35FGhx\n", - "tUn6SqcQQncnJw9NTuL/RuzQDliKko1PRRcvJRwgre4BhnmjhIP5Gfx121wHI+ub7cAX7eJ7UBfC\n", - "gN7nKRwV/9WEaNlOwTt4dXExccz6Yck4SDeMO+J3IaEgJ2k4JLR4XDCE70ry2/JXlbykguiaaRhF\n", - "c3keqo9SWDjNwOGyFo+LBlsA+ArG4/hUzj+A2YZRfL8O/uWNIC7WURaWKZt4Df89aupCPpDtmtEw\n", - "Xjzwc35BGs4nAX7gkmZOihY2jFFFGbhn7sLFJMo2jGZxCRgbfcdBomzDaOqUgMfNOjYmUa5hTBSX\n", - "gL9MFHITFK2yDaNZXAI+bNaxMYoWNIxGuTtcSX4fGRqZEEULGkajxB32XhhlfIiiOLiG0ah3h6tl\n", - "HXANY8aBC+z/ZZSxodEwspxMloKL5d4n8T/d2KAIICO4OgAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$- \\frac{\\sqrt{14}}{7}\\mathbf{i}_{1} + \\frac{3 \\sqrt{14}}{14}\\mathbf{j}_{1} - \\frac{\\sqrt{14}}{14}\\mathbf{k}_{1}$$" - ], - "text/plain": [ - " ____ \n", - "-╲╱ 14 \n", - "────────\n", - " 7 \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + ____\n", - "3⋅╲╱ 14 \n", - "────────\n", - " 14 \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m + ____ \n", - "-╲╱ 14 \n", - "────────\n", - " 14 \u001b[94m\u001b[1m1_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 15, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(r1 ^ v1).normalize()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "##### Ejercicio\n", - "\n", - "Usando directamente la fórmula para la derivada en ejes móviles:\n", - "\n", - "$$\\left(\\frac{\\operatorname{d}\\!\\mathbf{a}}{\\operatorname{d}\\!t}\\right)_1 = \\left(\\frac{\\operatorname{d}\\!\\mathbf{a}}{\\operatorname{d}\\!t}\\right)_0 + \\mathbf{\\omega}_{01}\\! \\times \\mathbf{a}$$\n", - "\n", - "Calcula la derivada del vector de posición $R \\mathbf{i}_0$, siendo $A_0$ un sistema de referencia que gira respecto al inercial con velocidad angular $\\mathbf{\\omega}_{01}=\\Omega \\mathbf{k}_0$. **¿Cuál es el módulo de la derivada?**" - ] - }, - { - "cell_type": "code", - "execution_count": 16, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "R, Omega = symbols('R, Omega', positive=True)\n", - "A0 = IJKReferenceFrame('0')" - ] - }, - { - "cell_type": "code", - "execution_count": 17, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAB8AAAASBAMAAABRHhjUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqu7IjJ23e9UmRCJ\n", - "zUQBtxRlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxUlEQVQYGU3OPw4BQRTH8W+w/i0ZUSkkJgo1\n", - "jSjdQK0yiU6johRHEBfY7Uk4gkrLDTiCf6tRjNndjPWq9z55b+YHtnor21V1s9mhoM8WeMKgndHS\n", - "gvOGVMDQzmQDyL1+I6SX4PkiORGSwpzMRdqlUWt8N/1Okj0eTMN0T60fQZ11CD0QfgQzxN7Aw8SQ\n", - "Ibgv8iauYx7wog3nQ65CFOOkXLMRgoLyDRpqGII5URQvjwrFrhJauwHiyn8t4l8SmsQ5EihtN3wB\n", - "oj40JUWOI5EAAAAASUVORK5CYII=\n" - ], - "text/latex": [ - "$$R\\mathbf{i}_{0}$$" - ], - "text/plain": [ - "R \u001b[94m\u001b[1m0_x\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "a = R * A0.i\n", - "a" - ] - }, - { - "cell_type": "code", - "execution_count": 18, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAACQAAAASBAMAAADBDWEbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\n", - "EO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6ElEQVQYGVXOP04CQRTH8e+wbpbd1WQqWzEe\n", - "QCsLGr2BVNByBE7AEi6wXoDEGENlosbGW8gJxGinFSCb9Q8ZfmOhyyvmvXwy8+YHmIN9iNyCv9pq\n", - "290O9CuUD+CkQfhP0VL3w1mV4i9RUlQpnYvqP54O3blm2C51BJ+enka/QlCox6XIvGlqNmHnXUP6\n", - "LDobaMNFTY9zC5lCLP2mtMdKAY7gWu0719ap5QVjWnBnCOeBs2QTHkgYYbqJiMdbT/fEwytql8ee\n", - "9goyyyuJK6m7ReRc68bNpj0+lKBa6an/caOCrs+1WePxZA2ClklZiRwIRAAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$\\Omega\\mathbf{k}_{0}$$" - ], - "text/plain": [ - "Ω \u001b[94m\u001b[1m0_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "omega01 = Omega * A0.k\n", - "omega01" - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\n", - "EO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABM0lEQVQYGVWPvUoDQRSFv0mU/VOylfWKdjaB\n", - "gEWapBFLrbQ0jZ3CvkEWXyApFZQsIqlsxEawiE/gbpfSvzKFaDQkmjDe2RSyF+Zw5ps7d84Aam2V\n", - "/7IG6XyzsOev7Iu1O5Oz8whHl+e8FUEtEF96QE0p6rp4sL5FFj9EmtLYCVk3FJwfEXckUguh5xtm\n", - "yvsUsacij1AYsaNj8bA0FilORGRtlFG9WLwgM8KRQ2tSSSLxSUy1CssD8d6zNMywXzJux4U2tOSp\n", - "ZiAD5czcTWIvZCYZy3At+1IbZZ5J4iefV5TahRsldyITx/Bmyh0uF6iGm8V3h6RzfotzckXhchMO\n", - "5BtDKzDc5w1Xj7H1kEN9CkeVbH7IlwzMVV/6vbrJky8dvgfFhsmfL739C91umqewdX+coT/7oGEp\n", - "9ULDIAAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$\\Omega R\\mathbf{j}_{0}$$" - ], - "text/plain": [ - "Ω⋅R \u001b[94m\u001b[1m0_y\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 19, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "da = omega01 ^ a\n", - "da" - ] - }, - { - "cell_type": "code", - "execution_count": 20, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAPBAMAAADuVgmRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmarid27\n", - "EO8D7p43AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0ElEQVQIHU2OsU7CUBSGv0tp2t5K2om5hpHF\n", - "pAkrr+AkIy5uDLyBjS9QR0kgTYiji08AT6A+gs5M2GCowdS/nbjJ/e+Xc/7/ngOYwSXnp3sT9ycq\n", - "+EW1WGaCXDJOBNEWcwLvIHb3kvsrKOYEv2L7IxnPYRMTfot9WXmDjhoXR7FTSXSHSjmNO1DVq9L3\n", - "TNzbScJPdf7wv8TksQYkyqrYuHGVe9EbPWKar4y5hlcjV9YOwbLC3Np2DVvyQfDwTGc9gqnWKb0E\n", - "Wx/x65K7+glmKf+/RDSDL80FuQAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$\\Omega R$$" - ], - "text/plain": [ - "Ω⋅R" - ] - }, - "execution_count": 20, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "da.magnitude()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "
Si no especificaste `positive=True` vas a ver algo como $\\sqrt{\\Omega^2 R^2}$. Debería haber una forma de simplificar esta expresión _a posteriori_, pero de momento no funciona del todo bien. Preparando este notebook nos hemos dado cuenta y ya les hemos avisado :) https://github.com/sympy/sympy/issues/8326\n", - "
" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Movimiento relativo" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "¿A quién no le gusta multiplicar matrices de rotación? Para esa minoría que lo detesta, existe SymPy. Para ello debemos especificar la orientación de nuestros sistemas de referencia usando el método `orient`, y recuperaremos la matriz de cosenos directores usando el método `dcm`." - ] - }, - { - "cell_type": "code", - "execution_count": 21, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "A1 = IJKReferenceFrame(\"1\")\n", - "A0 = IJKReferenceFrame(\"0\")" - ] - }, - { - "cell_type": "code", - "execution_count": 22, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAMwAAABLCAMAAAALSZigAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - "MquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAABL1JREFUaAXtW9mW\n", - "oyAQBUWmJ+7D///rVAEFakjCUn26eyY8RES91qUW9B4jpLGtEz+4bY6DENL0Ctrwg7mIFRl0BsnI\n", - "Yh5yDZdM2FvLIQLAsdOEu9aRmfZowWi7XREbOasIcOgV4+p+VD3dupLMoqMFjoxY4sjrnjT+qsup\n", - "xbgLJIfebHBAeFR5Rh1M0X6Oj2MXE7N3jxhZuOuM2KOfyDoys58KRLp5J2uLiyP1rRh3tyV4MC5Q\n", - "qsjctoO9wUno8dw2SYrz4xXluMaSuRl36zQZSKtxxBP0qNR4g45Ug1KUFmNPJoBN/TQ4P8VROnq/\n", - "9TDTjlbIZe4lFNRQCyJCJq421pKbcRBJMtMMRCT4XNrU2pXQtnqRQ3aqy/0gJugvrjoT13sONBJh\n", - "XHwsAC0mQ0FbjDu5OkLlJElmQbMGWH02G0OwDt1mjEqKqM0HFN58BUtGF7nElUy/30YYR2a3eWZo\n", - "bopxM8h45+Fq6qwG/pvZFU2gmDHwhFBoPRLsLQ0JhfFVCzCejHWmDxJRjqudZ56FGR0D5zirzQ7J\n", - "sxgXoGCvJ7NhwCKZxQZhDpkA48nYC69kCnCdScOTAnDvmX5Cr6yem/DhgOGBKTM5rFMtgtMTLcIk\n", - "yZTjbnY21mel2eYMZI2YfbG4DTZdOh/cPlEXyF5Mmd0Vt/V1AYgwSTLluMomXe/vnCwA04xWj1DN\n", - "ZlwOllEMtgDsPmtcwgt0BZD0XETvOCUcEoYizG6n1P2GMKvARWdqayXcJElG6K5XwAUiCBacDhLn\n", - "Nq5qtQsOGkYBJft1X0da/haXYHjCo0YwsjOm0/ZXDruZbTWswsWFMDzjpsk8MobG6cnOpowfnBge\n", - "Zxpx68isFFGYMr6NYSGnkfJtI24dGRGyJxisX6d/OPdxpw23kgy9REVvkBmPDc050oZbSeaYLNbI\n", - "Q7zlGP3wHFy3jq0It5bM8Ybfpv8m821ccTHk7ZnLhHyb3WbPNKl2iWlowcslw6baJew/DtFCg2P2\n", - "SV2EJ6/jaaFfJQLSa3ZA8Z1i1e4KcNkvxWMQAYMFxapduDLdKcXjEAGDJcWqXbgy3SnFc+94pSIg\n", - "l2qX5kCj9KJk913KQPeJupgjAhJ4EP/YVDtedTFHBIxcmFU7bnUxQzeLXASzasetLj4ho5cttMW/\n", - "2bOqdvcaVqO6mBABf338Pjjj3GVV7fjVxbMI+OcDVVgSWM5EYI9Xtbv3TKu6mCECRk7Mqh27upgh\n", - "Ah7IBPGPRbXjVxczRMDAhlu1Y1cXGUTAQLZRtQs41GnDy30FoLtdto2q3QUN1OImdbGRDLsaSPJb\n", - "eDATBepiKxl6meJSA5vwWsmwq4EtKmAzmbuw/8KBN5kvnPynt3575un0fOHB/9kzJ82N0wcNwFFq\n", - "K/TMSXPjJFMNDFJL+CqxjMxZc2MkUwus976rJXPW3BjJNACrWjJnzY2RTANwLZmL5sZHpgW4lsxF\n", - "puIj0wL8JgN+uGhufJ5pAa71jDhrbnxkWoCryZw1N0YyDcDVZM6aGyOZBuBqMvbrzPDhHSOZBuB6\n", - "MifNjZNMLXDfzWajb0fLns04zf8ErDeZT5hUFsi3Z1im8RNA/kHP/EN/OtX4h02lXn8w/gmRwQVp\n", - "/3SqlPgLJN1NThRRO9AAAAAASUVORK5CYII=\n" - ], - "text/latex": [ - "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )} & \\sin{\\left (\\phi \\right )} & 0\\\\- \\sin{\\left (\\phi \\right )} & \\cos{\\left (\\phi \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" - ], - "text/plain": [ - "⎡cos(φ) sin(φ) 0⎤\n", - "⎢ ⎥\n", - "⎢-sin(φ) cos(φ) 0⎥\n", - "⎢ ⎥\n", - "⎣ 0 0 1⎦" - ] - }, - "execution_count": 22, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "phi = symbols('phi')\n", - "A0.orient(A1, 'Axis', [phi, A1.z]) # Rotación phi alrededor del eje A1.z\n", - "A0.dcm(A1) # \"Direct Cosine Matrix\"" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Usando el argumento `Axis` hemos especificado que rotamos el sistema un ángulo especificado alrededor de un eje. Otros métodos son:\n", - "\n", - "* `Body`: se especifican los tres ángulos de Euler.\n", - "* `Space`: igual que `Body`, pero las rotaciones se aplican en orden inverso.\n", - "* `Quaternion`: utilizando cuaternios, rotación alrededor de un vector unitario $\\lambda$ una cantidad $\\theta$." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "
¿Qué es lo bueno de usar uno de estos métodos? ¡Que **siempre** tenemos la transformación bien definida! Es imposible meter \"a capón\" una matriz de rotación que sea incorrecta o absurda.
" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Diferente sistema de referencia\n", - "\n", - "Para expresar un vector en otro sistema de referencia, no hay más que usar los métodos `express` o `to_matrix`:" - ] - }, - { - "cell_type": "code", - "execution_count": 23, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAALMAAAAVBAMAAAD/f8IzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJVCLvMs1Edmar\n", - "3bti/yyrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACnklEQVQ4Eb2STWgTQRTH/2kybto0MSgqiGhE\n", - "e/EDFlSs1cOCKNiD3YsiFKwt9aKIOVWtFYNeggddUKwg0kAVxA8IShG8GE8evARFb9JFUKQVLX5h\n", - "bXR9b7KbzG420pMP8vY/b97vP5OZAf5nnObFBPBILpoanMfafkQ7lg5nhM31KNCeZdHzk3NjaEfq\n", - "tQDS7hTqc6qKyyXJGjku91U4N4b2tV4LIG2OVZ9T1bgcLKcslWYok0sUrcgAgj3KnCoPy8Eo5aVq\n", - "Weruhoos/AtRiNayHOiU4wYlRz0QxVqckH2cgsiYwzTFok2bEekatzDcfZuG0QwlLduxy5UHQqy1\n", - "m3dyBys4OTB8rxCCiMmqtRjEkHnGxFTERpL6UkVKa0UxbiKRI9lC1ppNgsPd9UqgjI/AxYL4TtUg\n", - "gs86NJv+dQ492ZfA2VP9/OKQNLg3YUZLiJVpTNbaBhYcrvXVNHS2fg/MUTWIkLVEVmVoV9+Ahfrk\n", - "lEl9CyzgPi3SUoY2Q2PetVyDtGed/HMLbP0B+BWC8K4ZGSsQOkvW9uJLX9y+F2Qdt4PWnfn863z+\n", - "HPVEOp0MW3+qW6uIZ13bdYeJJ7Rj/ndz6EWfUb35sF3vQ2JWsQ4injWdNYwB2v0VHe3F6p1cwATe\n", - "AgmaCj2QncBzxZqvUUU8a3EUUXMoi/OxfqRMenEl4Ab0VIZc6RdqvTuNd3wa7oEEEc8aK9ZtgViz\n", - "w2rdfncbWfH719ZPbyUZtyg5zkzDNV5+/NAacZ6OOM/2/ig1IjVrwn3xikaiyKX9Xj34Qry69/Uj\n", - "y5paPyAgYTLV5aGxQ6667lX8Xz/iZI8bqCFqpzxwLkRybllsrBRcGf7xI87ob4Qjwgba2IKB+YUf\n", - "ufZmdTOs152YaNbQWA9F/gLES9pSGqLHmQAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$\\operatorname{cos}\\left(\\phi\\right)\\mathbf{i}_{1} + \\operatorname{sin}\\left(\\phi\\right)\\mathbf{j}_{1}$$" - ], - "text/plain": [ - "cos(φ) \u001b[94m\u001b[1m1_x\u001b[0;0m\u001b[0;0m + sin(φ) \u001b[94m\u001b[1m1_y\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 23, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A0.x.express(A1)" - ] - }, - { - "cell_type": "code", - "execution_count": 24, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAEwAAABLCAMAAADtTJGhAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - "MquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAA0dJREFUWAntmNuW\n", - "nCAQRQtBMhEVCf//r6kq7g62MtNZeRkeuhFlWzc5IgjPbYJvtCMwAISXCtv8DRZYIkyeYOIxR9h8\n", - "6Uo9W0+1Y7DVZRYY7k4VbRC26zMM9jIyBlPBGp6uVaBUY2OwjeMUIEv0T2/ZtCHYcuR5EEOGA3su\n", - "gxampTGGzmmjlFmwI9SsVAqLkQkmBMh1DnaW0Qa2bggS6Is46DqnQHP2kkEu1YWcYcX+Hqoj3Qsa\n", - "2E7DM1bdwYHG+ls2Sl8K+xEdIqhFkOGHZk73amDaRzeED7O8gcM7FbxB6EaOAyiaTTeQjBFYqqHV\n", - "li0+Znv2YZZ3GLzdp3sk2EH3JNjOQejDPlsmV7LKRjZENz16SSFbgwMlx7VlwDHDqMHG/i5+mTlc\n", - "E86kFhOwowMUMheiYvsJWDeaZTCbG1XkbmDmBLgYtRBwIFPwJpEFMkaaXKhXDT1JhSz0AAtuwsAt\n", - "xirLBYeDTKF/Ia2zJj3hewgwjrcwuvJV4/KjCyhksa1ffJzAJo8oZLGZWAJ4OGYZ5OglFOgc/mFY\n", - "WhyLNQlP9EHL6mCxcZW/47DsXq8zalmPkcd+YDkUVeelBl+UhthKIVUoSGVGY7yeQK3BVzBcY3vt\n", - 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"source": [ - "#### Símbolos dinámicos\n", - "\n", - "Si queremos especificar que un símbolo puede variar con el tiempo, hay que usar la función `dynamicsymbols`:" - ] - }, - { - "cell_type": "code", - "execution_count": 25, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "from sympy.physics.mechanics import dynamicsymbols" - ] - }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAACMAAAAVBAMAAAA+1EraAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiZl2RCLdEO9Uu81m\n", - "qzKufkATAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVQYGX2QsU7CUBSGvxZKqUIFHXQhdncQ\n", - "eIJObiZMhDDhwMDWxMXNmOjOG0A0bhq7uXZhI4RH4BE6kLj6t1cpEyc59/znO+fce3LhgPlJVlzt\n", - "dwzzpL7ZY9/gjiEuUKkBXgqdAlV7YKvloUA3khV1Hk8lTrthdsLt12cbK4B+6L44Uz6EZ/Kahp/g\n", - "tRQxUXol90bYP5IWPCu9k/sp1TlcnOfI3Rr0qFcu9YoGyykR5REniZCqut6KvU12vb2mdu1HtLTp\n", - "+l5HAM1lYp+BVnXe2lo11IixSs/EwT/QioHRGtjZe67ceAfAfOHR33xecJIsLOAXOGcpup/x+cEA\n", - "AAAASUVORK5CYII=\n" - ], - "text/latex": [ - "$$\\alpha{\\left (t \\right )}$$" - ], - "text/plain": [ - "α(t)" - ] - }, - "execution_count": 26, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "alpha = dynamicsymbols('alpha')\n", - "alpha" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Y pedir su derivada con el método `diff`:" - ] - }, - { - "cell_type": "code", - "execution_count": 27, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAADcAAAArBAMAAAAu4Q2VAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARLvvmVQQid3NIqt2\n", - "MmaorGxOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABr0lEQVQ4EX2TvUvDQBjGn7TNpY22FlwEhwpO\n", - "fqDo4iZxEqlg/wFpoYuD0AwOih/UQQRdOohzqDi5iItLh7jootDJuYOToETBuqnv5dr0KHe+cHfP\n", - "vb9c7u5NHoCHMe6Eo7obKqjzYTZX+gdu/8NwpoX27dOXFg57sbYOskUkajqYaCPp6mB8DJWsDlZc\n", - "FFHS0EoBE6angSk/c2ppGFhj/+pOhukWn73JqZ6uh3Kg2ctI6hxgJ4Dy7oN0aTMAHqTnIxn3EJas\n", - "HGUkMUKalyxVkJKAtbpE8y1gdHbKh+XI0D5E2bvEDOWK1DI1GcZdxJx7HFFuhZpZpS6KtRIdxMUc\n", - "JY6ppQP8diPAPO26UOKQfQgYrQP7Jkj/C73WCOgzGdUeA34I0uXpQFbNbPYdKA+8BM/YpRpcb1Ln\n", - "yCutg3cs+6Ai2NM+FSErw45OekJchEOfjTKOgHtiIBtteELyfjKUrFMgshGvVzfEx455Yk424vXq\n", - "ht3i6rUzJRvxeimC28i4+WwoEBDaiP8WihA20jhJ2EjjJGEjjZOEjXJgii0hbLSDdRUUNsozXwWF\n", - "jeqPKvYHoMNlfCAqdJQAAAAASUVORK5CYII=\n" - ], - "text/latex": [ - "$$\\frac{d}{d t} \\alpha{\\left (t \\right )}$$" - ], - "text/plain": [ - "d \n", - "──(α(t))\n", - "dt " - ] - }, - "execution_count": 27, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "alpha.diff()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "##### Ejercicio\n", - "\n", - "![Sistemas de ejes asociados a la pala](../images/ejes_pala.jpg)\n", - "\n", - "(Sacado de Cuerva et al. \"Teoría de los Helicópteros\")\n", - "\n", - "**Obtener la matriz de rotación de la pala $B$ respecto a los ejes $A1$.**" - ] - }, - { - "cell_type": "code", - "execution_count": 28, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "A = IJKReferenceFrame(\"A\")" - ] - }, - { - "cell_type": "code", - "execution_count": 29, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAQAAAABLCAMAAACoRKlPAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - "MquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAABq5JREFUeAHtXOvW\n", - "qygMxeuc8VrH93/XSQIJoGAVadf51qc/CkbIJjFEZLdVxUpHqX7Z8dJ2K1WsVQ1H88vsVzNaXa7o\n", - "gOK67cVs+9jujnRmqVRsh5Sao1qxatDjiBmIy1Moc6IDhsVR33Pdk5Y8TKlws6OyGOvgZU+1ACpP\n", - "zEBcBhWpturrisemUh0wtY52GY+Rtq8Br07cRCosOCiLVbR5rcKAKox4CDjBXDcjBIREB9TuKAee\n", - "DSxtIKzg4FNbIXHSh+jC3gIomjeIXusN3jyioGcfJTpgpFtsVM98wtJKa28JChpJxXRIKFg1dRVA\n", - "xeIN4hHgQk+7ZjUhnOaA7uXawNEg0peRYLDRIRW3W7g+FDI9nQaimmQMqES8RTwAXMkB3WrGFnYA\n", - "5Im+xxZtX9d9B5Wibuqaw0b1FQ2k7SvMWDCeGj2ipV25rEtJmcw04yvUJ/ZhAIYFB1hMY1XAI0ry\n", - 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"AABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$\\left[\\begin{matrix}\\cos{\\left (\\psi{\\left (t \\right )} \\right )} & \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" - ], - "text/plain": [ - "⎡cos(ψ(t)) sin(ψ(t)) 0⎤\n", - "⎢ ⎥\n", - "⎢-sin(ψ(t)) cos(ψ(t)) 0⎥\n", - "⎢ ⎥\n", - "⎣ 0 0 1⎦" - ] - }, - "execution_count": 29, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A1 = IJKReferenceFrame(\"A1\")\n", - "psi = dynamicsymbols('psi')\n", - "A1.orient(A, 'Axis', [psi, A.z])\n", - "A1.dcm(A) # T_{A1A}" - ] - }, - { - "cell_type": "code", - "execution_count": 30, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAP0AAABLCAMAAACMeHGmAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - 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"⎡cos(β(t)) 0 sin(β(t))⎤\n", - "⎢ ⎥\n", - "⎢ 0 1 0 ⎥\n", - "⎢ ⎥\n", - "⎣-sin(β(t)) 0 cos(β(t))⎦" - ] - }, - "execution_count": 30, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A2 = IJKReferenceFrame(\"A2\")\n", - "beta = dynamicsymbols('beta')\n", - "A2.orient(A1, 'Axis', [beta, -A1.y])\n", - "A2.dcm(A1) # T_{A2A1}" - ] - }, - { - "cell_type": "code", - "execution_count": 31, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAggAAABLCAMAAAAicY+6AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - "MquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAADAVJREFUeAHtXeu6\n", - "qyoSxMQ4Z3LP+P7vOghdDSJIQzDHfCv5sSEtVtFNB9FFbVU3ms9B/T5/MgJPO/5KdeOx15/Tn4zC\n", - "z2n1mEb/ME6J0JWHo3vgnOvjcaW6M6oHYaJE60zJCIA07dmqGNdVMoiZwww9I1RsdoyFniR5HfSe\n", - "IveoTITrC47qmWR42VF3Rn3wQJmAEu1XS4dw8do5q4fLBF7DdLW79dGDDtonVM7sMRZ5ovKMu4pc\n", - 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"⎡cos(β(t))⋅cos(ζ(t)) sin(ζ(t)) sin(β(t))⋅cos(ζ(t)) ⎤\n", - "⎢ ⎥\n", - "⎢-sin(ζ(t))⋅cos(β(t)) cos(ζ(t)) -sin(β(t))⋅sin(ζ(t))⎥\n", - "⎢ ⎥\n", - "⎣ -sin(β(t)) 0 cos(β(t)) ⎦" - ] - }, - "execution_count": 31, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A3 = IJKReferenceFrame(\"A3\")\n", - "zeta = dynamicsymbols('zeta')\n", - "A3.orient(A2, 'Axis', [zeta, A2.z])\n", - "A3.dcm(A1) # T_{A3A1}" - ] - }, - { - "cell_type": "code", - "execution_count": 32, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - "MquZdlQQQOkwRM0iu+/dZolsKH0JNQAAAAlwSFlzAAAOxAAADsQBlSsOGwAABxdJREFUeAHtXNmW\n", - "qzgMNOu9k7AO//+vI8mWvOFg05Dp0508xI5iVC68AFXpVtVGr1r9ktes+SpVbU0Lr+6X8FYLsq03\n", - "JF79Fs6W5xISf/b2y6/U+mZtm/ed0GK4gPgwbhcRf8LK6efhK+eu5NhiOI94Pzb1RcSXCbu9Pks6\n", - "/4W25XAecUBuLyI+0jWiuyjb4Skph7uL+EbEH9ubLhXlcDcR77cGR+mxtYeDdUWDE3A3ER+2FQlV\n", - 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"theta = dynamicsymbols('theta')\n", - "B.orient(A3, 'Axis', [theta, A3.x])\n", - "B.dcm(A3) # T_{BA3}" - ] - }, - { - "cell_type": "code", - "execution_count": 33, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAgEAAABLCAMAAADeZKRwAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - "MquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAADUVJREFUeAHtXeu6\n", - "qjgMRUXmjHeH93/XKW1WmpZeQkW259v6w5RYkqw0QMEu7Xajfe277+t3ZeDhBr7rduOhN6/j74L/\n", - "Rdvdp2Hfj1MF7N6Zjt2drF8CL6zuujv5hwz6fdoGxx3AYa1Hw7A+DUEYz/3tFXB5wuMZjUl6tdnY\n", - "UwlAyo4/1N5d+6RnH7eE47UCDcNKGnqTcjic+8OSI/r9FXAbgHXf9zgddB2ph4c9lG7UBxK7/Jzc\n", - "jXKEfRxpOGk03Q/AuZmLOSXVB11qvb0CeiRy2JvZBs81oD6ac5B5YROyFPKPfsYBBnCgjdAwrO1C\n", - "vl8nX+cFpff2CrjS5XK4nWQeoD64WAcbuKlekrLrR7URdwgH2gjND8B52ju648gn3mr23l0BpweF\n", - "cPMXAKNh9YNOEdPJa3pBuq0ffL/sUhdTjjuAw9oYzfZwRlsBp5FPttUUvlwBZuZxPk/+hnPfn6fj\n", - 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\\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" - ], - "text/plain": [ - "⎡ cos(ζ(t)) sin(ζ(t)) 0 ⎤\n", - "⎢ ⎥\n", - "⎢-sin(ζ(t))⋅cos(θ(t)) cos(θ(t))⋅cos(ζ(t)) sin(θ(t))⎥\n", - "⎢ ⎥\n", - "⎣sin(θ(t))⋅sin(ζ(t)) -sin(θ(t))⋅cos(ζ(t)) cos(θ(t))⎦" - ] - }, - "execution_count": 33, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "B.dcm(A2)" - ] - }, - { - "cell_type": "code", - "execution_count": 34, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAABJQAAABMBAMAAADJpvAnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMquZdlQQ3SJEie/N\n", - "Zrv3ZvUrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAXvElEQVR4Ae1df4hl51n+Znbnzr2z88tSibKG\n", - "maRFKKIZzSKh7R+XdgK6pDhiIZCAGdutLck2O1LENVb2Sv8Rll0ngggu7V6SqO0SYWs1bEDxoqRB\n", - "neqE5o/QpjguiwRqm82mkW7UjM/zfud8v88vZ+YMhPvBOfc733m/732e93vnnO+ePc9etbS7e0uN\n", - 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\\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" - ], - "text/plain": [ - "⎡ cos(β(t))⋅cos(ζ(t)) sin(ζ(t)) \n", - "⎢ \n", - "⎢-sin(β(t))⋅sin(θ(t)) - sin(ζ(t))⋅cos(β(t))⋅cos(θ(t)) cos(θ(t))⋅cos(ζ(t)) -\n", - "⎢ \n", - "⎣-sin(β(t))⋅cos(θ(t)) + sin(θ(t))⋅sin(ζ(t))⋅cos(β(t)) -sin(θ(t))⋅cos(ζ(t)) s\n", - "\n", - " sin(β(t))⋅cos(ζ(t)) ⎤\n", - " ⎥\n", - "sin(β(t))⋅sin(ζ(t))⋅cos(θ(t)) + sin(θ(t))⋅cos(β(t))⎥\n", - " ⎥\n", - "in(β(t))⋅sin(θ(t))⋅sin(ζ(t)) + cos(β(t))⋅cos(θ(t)) ⎦" - ] - }, - "execution_count": 34, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "B.dcm(A1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Velocidad angular\n", - "\n", - "También podemos hallar la velocidad angular de un sistema respecto a otro usando el método `ang_vel_in`:" - ] - }, - { - "cell_type": "code", - "execution_count": 35, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAIYAAAAaBAMAAACA8zmaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - 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\\operatorname{cos}\\left(\\psi\\right) \\dot{\\beta})\\mathbf{j}_{A} + (\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\theta} + \\operatorname{cos}\\left(\\beta\\right) \\dot{\\zeta\\left(t\\right)} + \\dot{\\psi})\\mathbf{k}_{A}$$" - ], - "text/plain": [ - "((-sin(ψ)⋅sin(ζ) + cos(β)⋅cos(ψ)⋅cos(ζ))⋅θ̇ - sin(β)⋅cos(ψ)⋅ζ̇ + sin(ψ)⋅β̇) \u001b[94m\u001b[1ma_x\u001b[0;0m\u001b[0;0m + ((sin(ψ)⋅cos(β)⋅cos(ζ) + sin(ζ)⋅cos(ψ))⋅θ̇ - sin(β)⋅sin(ψ)⋅ζ̇ - cos(ψ)⋅β̇) \u001b[94m\u001b[1ma_y\u001b[0;0m\u001b[0;0m + (sin(β)⋅cos(ζ)⋅θ̇ + cos(β)⋅ζ̇ + ψ̇) \u001b[94m\u001b[1ma_z\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 37, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "B.ang_vel_in(A).express(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Derivada en ejes móviles\n", - "\n", - "Hacer una derivada con la fórmula lo hace cualquiera, pero SymPy puede encargarse automáticamente." - ] - }, - { - "cell_type": "code", - "execution_count": 38, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAABoAAAASBAMAAAC3N9OQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu+9UzSKrMplmRBCJ\n", - "3XZModtBAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNgEDJgQADm/w4IDgPr/wQkHkMx\n", - "MgeVzY+iktUfRV8/mMcYwMBQCNQF4c1fwFCRC+Nx909gYLCF8Xj2NiDx6hgFkHgJHBcQPE4Dvg9g\n", - "Hv///wwMezsaf0PkgAYxtDKwf4fzmIHG/2RgMAJJMDDICzAE/p0QcT8PAHetIcptQTo1AAAAAElF\n", - "TkSuQmCC\n" - ], - "text/latex": [ - "$$\\mathbf{i}_{A1}$$" - ], - "text/plain": [ - "\u001b[94m\u001b[1ma1_x\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 38, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "v1 = A1.x\n", - "v1" - ] - }, - { - "cell_type": "code", - "execution_count": 39, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAPAAAAAZBAMAAAD9ML7rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - 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], - "text/plain": [ - "⎡ 0 ⎤\n", - "⎢ ⎥\n", - "⎢ 2 d 2 d ⎥\n", - "⎢sin (ψ(t))⋅──(ψ(t)) + cos (ψ(t))⋅──(ψ(t))⎥\n", - "⎢ dt dt ⎥\n", - "⎢ ⎥\n", - "⎣ 0 ⎦" - ] - }, - "execution_count": 40, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "dv1.to_matrix(A1)" - ] - }, - { - "cell_type": "code", - "execution_count": 41, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAADgAAAArBAMAAADf6lYYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARLvvmVQQid3NIqt2\n", - "MmaorGxOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAB1UlEQVQ4EX2RP0gcQRTGv70/M95xrhIbuxNS\n", - "JUU8xMpCjpAUwcJNJwFxIE3Ewi0sFBSuE7RR0C7FErASgqRJc8JaCZJAaotoISGmugTuUup7b+5k\n", - "znV8MO+97/1md2fnAzieTErxpLZnzuOg8wjUjUdgqe6Fh+sXkQ8GUxg3Pvj+Em98DLMGm144AfXX\n", - "B9V/FFvKQ1UbuZ13HoiP+H0U++DS6XlzuAsHpf7kHP7ozu6K5q48JvpYspMElgzwIUHJmUv7mfMp\n", - "rQWgcv8gfxhO0aI7UzUWTvDmoEGJ72yfVi/USoq40oCuI/jyrwms9AhVXWyVo0oHuUvQndHg2oEH\n", - "hUZoUMNABBR2CMw50ORj+pM68sPAQEzgAje9aKGaaKhU4DJtIOjEKg6gjby2CvJp1GF4iV94CznQ\n", - "Gub7D4QRpCG9Th8BMyoFXrlPFl+/OCPNl/DpGzW7LsRgJHLLDss1rsHTupUVI/XKKp1KHYrYIvqa\n", - "KOS4BxYlo2rYIjLJyq7Zz61aFYtsz1lcDRM72LMWWdGXy1+/t61FfWMrRpJcx1qUhWqa/RGLsrDQ\n", - "YX/EoizMj4H8EYuycDmmfzRi0QMwwrNiIhZlYSkNt7W1KAtVc+PwxFqUgbfaSGivipnJwwAAAABJ\n", - "RU5ErkJggg==\n" - ], - "text/latex": [ - "$$\\frac{d}{d t} \\psi{\\left (t \\right )}$$" - ], - "text/plain": [ - "d \n", - "──(ψ(t))\n", - "dt " - ] - }, - "execution_count": 41, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(dv1 & A1.j).simplify()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Puntos, velocidades y la rueda que no desliza" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "El último paso que nos queda para completar la cinemática es la posibilidad de definir puntos en sólidos y aplicar su campo de velocidades. SymPy también permite esto, y para ello no tenemos más que importar la clase `Point`." - ] - }, - { - "cell_type": "code", - "execution_count": 42, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "from sympy.physics.mechanics import Point" - ] - }, - { - "cell_type": "code", - "execution_count": 43, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "O = Point(\"O\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Para trabajar como lo haríamos en la escuela, vamos a especificar que $O$ es el origen de $A$, y para eso vamos a imponer que su velocidad es cero con el método `set_vel`:" - ] - }, - { - "cell_type": "code", - "execution_count": 44, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "O.set_vel(A, 0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Para definir nuevos puntos, podemos utilizar el método `locate_new`:" - ] - }, - { - "cell_type": "code", - "execution_count": 45, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "e_b = symbols('e_b')\n", - "E_b = O.locatenew('E_b', e_b * A1.x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Y para obtener vectores de un punto a otro, el método `pos_from`:" - ] - }, - { - "cell_type": "code", - "execution_count": 46, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAACwAAAASBAMAAADS2iHvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\n", - "zbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVQYGWNggIMXE+BMJAbn/wtIPDiT9f8G\n", - "OBuZsRKZQ4jNj2oIo7JJAEgL630Us9MDWBXY7gLF/cHCfEA164CKOhlEFjCowIXPJzAs2c7AwN1j\n", - "fJCB4SZMmNH/AAPDdAYGfrDWrzBhJnEHiLACUITts7UAxOzFfA/AwhxAYQPmQmYHiPAGlgKwMPMF\n", - "BpEAPgOeC2BhngnMH8DCDCY2Bxg4Elg28P//z8Ag7uL3ESIMNJqBX4BjAohmcGXg+oUQZhc4Bxbl\n", - "PMDA8IeBYQaYw8DA5aQAZr1/wBD3/UBI/WYAfoo7IFK/ix4AAAAASUVORK5CYII=\n" - ], - "text/latex": [ - "$$e_{b}\\mathbf{i}_{A1}$$" - ], - "text/plain": [ - "e_b \u001b[94m\u001b[1ma1_x\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 46, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "E_b.pos_from(O)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "
La notación de este paquete está influenciada por el libro Kane, T. R. & Levinson, D. A. \"Dynamics, Theory and Applications\". Es ligeramente distinto a como estudiamos nosotros en la escuela, pero ¡están abiertos a que les hagamos cualquier tipo de sugerencia! https://github.com/sympy/sympy/issues/2584#issuecomment-31552654
" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Por último, el **campo de velocidades de un sólido rígido** se formula usando el método `v2pt_theory`.\n", - "\n", - "$$v^P_A = v^O_A + \\omega_{A_1 A} \\times \\mathbf{OP}$$\n", - "\n", - "Este método pertenece *al punto del cual queremos conocer la velocidad* y recibe tres parámetros:\n", - "\n", - "* `O`, punto de velocidad conocida respecto a A\n", - "* `A`, sistema de referencia donde queremos calcular la velocidad\n", - "* `A1`, sistema de referencia donde están fijos ambos puntos (_sistema de arrastre_)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Por tanto, para hallar la velocidad del punto que acabamos de crear:" - ] - }, - { - "cell_type": "code", - "execution_count": 47, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAADoAAAAYBAMAAABZ8fMmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\n", - "zbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABUUlEQVQoFb3Rv0vDQBQH8G9yaZrEkN5Q6GjA\n", - "gmvAOilYRAeXmkGQgkg23UzBXVfBoZM6FnRwsgEHR9PBVbOJg9A/QLC1WLEI+pLzRy3E0QcJ997n\n", - "3T2OA4YjGwxno+vxaLTyP7lBY05rabNkgsveXzrXT9M1At1L06s0SOo2oL4/DbdIEyU3zqWiA5v5\n", - "WP2lG27GVu8BWe9k6qwLK1GLdpwBmT3kGygCVcU3OXY/9cbDyQUwtj/VAu4Artl03bJQaTkEDoFc\n", - "8mjPNHjLlSE5QuVCWahNoPZmOCZRhcyFHlvtRDVSh9VYGRW0cAChgeInyiLkXcsxI2zCMT2hZp11\n", - "EkVpNoTmKQH0pYdpmhLfqLBQ6QqlAnJcqwMGfVIj1kUYgx/N8mvqYZz6Btor9BB4A47ijRTGvE3/\n", - "+HVz/aaPxzbWX8KVnXMqfAejlXq7TSd8xQf1n06KqQYDUwAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$e_{b} \\dot{\\psi}\\mathbf{j}_{A1}$$" - ], - "text/plain": [ - "e_b⋅ψ̇ \u001b[94m\u001b[1ma1_y\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 47, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "E_b.v2pt_theory(O, A, A1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "##### Ejercicio\n", - "\n", - "![Disco que rueda](../images/disco.png)\n", - "\n", - "(Apuntes de Óscar López Rebollal)\n", - "\n", - "**¡Halla la velocidad y la aceleración de $P$!**" - ] - }, - { - "cell_type": "code", - "execution_count": 48, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "# Creamos nuestros sistemas de referencia\n", - "A1 = IJKReferenceFrame('1')\n", - "A0 = IJKReferenceFrame('0')\n", - "A2 = IJKReferenceFrame('2')" - ] - }, - { - "cell_type": "code", - "execution_count": 49, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAaBAMAAACgM6QiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMkS7zRCZdiKJ71Rm\n", - "q90icBAQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACGUlEQVQ4EZWTsWsTcRTHv3fmkktzJKVjsST/\n", - "gCYggm5FcCrCDdWp2AzVTMJNDRXEgIOLYAZxKKiHUzfTpUtBbxARQdrSoUMpRlBRENTaVEtA/L47\n", - "k/5enO7Be/fefb5ffr+7+x2ACaSMQPRWLaULp8SQ8VkuS3dSijcuVcdcxPmY5mVsMPGa6SbLbsms\n", - "wj67pulHjj1KnEdLfrzs1RYyyiLDpF/SdBHI1wk2Gk1gic0LINfmVcVXlDqK3gfGOkDhl8hWmAtc\n", - 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}, - "outputs": [], - "source": [ - "# Orientamos los sistemas de referencia\n", - "A0.orient(A1, 'Axis', [0, A1.k]) # A0 no gira respecto a A1\n", - "A2.orient(A0, 'Axis', [theta, A0.k])" - ] - }, - { - "cell_type": "code", - "execution_count": 51, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\n", - "MquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAABmNJREFUeAHtXOnW\n", - "qyoMxfGe41D1+v7veiCwIzJYUdrVb331R6ERsrMJg03aimKlqxS/5Jo1XyGKtarl1fwS3mJSbMtV\n", - "ES/SOReT7tNOpiIgkfIJCrmSDmD3gOoNLISWBDZdJD4s2rBuaZuKqpDQmxLMuWLziNWLsQ7eguoN\n", - "TEC0QzsGa6u+rmCXuEr80RJkMcpypbqRtPOg7jzotl2B4KAs1j541wcTYTRGDal5yLVsrJO3LxKv\n", - "jYWzmuer2h0gaeQUst6ynKTXXqB6A2OtDhpahnCmUUl7jM1F4iO5VWhlq2JvJKLSmluCkXKuKNhr\n", - "lw8WQzsCW+jUavT0vOrxbtYUZrW8C+VxSMRs5oKaWHRxxbw/KIaCl6DVCqo3sDjaAdhKxDuanlJ7\n", - 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"text/latex": [ - "$$\\left[\\begin{matrix}\\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" - ], - "text/plain": [ - "⎡cos(θ(t)) sin(θ(t)) 0⎤\n", - "⎢ ⎥\n", - "⎢-sin(θ(t)) cos(θ(t)) 0⎥\n", - "⎢ ⎥\n", - "⎣ 0 0 1⎦" - ] - }, - "execution_count": 51, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A2.dcm(A1)" - ] - }, - { - "cell_type": "code", - "execution_count": 52, - "metadata": { - "collapsed": false - }, - "outputs": [], - "source": [ - "# Creamos el punto C, centro del disco, y especificamos su velocidad\n", - "# respecto a A1\n", - "C = Point('C')\n", - "C.set_vel(A1, xi.diff() * A1.x)" - ] - }, - { - "cell_type": "code", - "execution_count": 53, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\n", - "iUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVQYGWXOMWoCQRTG8T84q6siLuIBUthZ\n", - "WHgA9wbuEdJES2MrgnuEba0SsEkXwVIIamPrASxEbEUlsCpIJrODMDi+avi9b957YJUYeY/idOJW\n", - "OyAto0en+I57Jis3ljdVsOPzYzEfPjRCW6EH+RsHubJaMUwi3IblIh4OAhUdrFj3PfMne8L50l4I\n", - "0y+U6klVIPMJN+2pUFxMvviKq1aoObmN+2e8GSTn6PnkT8bV+YUrXrKXVGS8Bs5VjLWvDS/kGyyH\n", - "eo4zM35/lZP8np3dkP73WFSn3Sff/ZKT8mj7tj7X9A8NW0cG87ppKwAAAABJRU5ErkJggg==\n" - ], - "text/latex": [ - "$$- R\\mathbf{j}_{2}$$" - ], - "text/plain": [ - "- R \u001b[94m\u001b[1m2_y\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 53, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Localizamos el punto P, punto fijo del disco, respecto a C, en\n", - "# el sistema A2 (que gira solidariamente con el disco)\n", - "R = symbols('R')\n", - "P = C.locatenew('P', -R * A2.j)\n", - "P.pos_from(C)" - ] - }, - { - "cell_type": "code", - "execution_count": 54, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAR4AAAAZBAMAAAAVlVzpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\n", - "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMmYiu80QdonvRN2Z\n", - "VKvu110NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEqUlEQVRIDe1WS2hjVRj+btMkbdI0wY1UGRud\n", - "1YA4URSZVYOC1VWyiSgKk1EQhdEEUdqFOBdFxkftFMUKo3WuoqtBOihSxcdEBKu77AZUaHaDm7Q+\n", - "Z+h0Gr//P+fc3qRJZ9y58Iee8/2P8/1fz+O2wL+ydKu3fHekt+Iq/G+uoqZvSa3cG94d6a24sp/4\n", - "68o1/1f8R3fAy+8l7BomH8njRltD2GOPQZPDSz3xPV0vDyRuMitZKNB7p0pEu50/t3Ta7bvU6xli\n", - "RSB1zy8Yz2lCYI/thybX+l7JKO9EfWel9PzwyYajVdix+ful7E9gqiigx+JVYKKarcK8K4VdJYn5\n", - "WU1ObXeFnRPhnfRdEGBPbxF/w9AqTHYCzSelzLvA3/I39buHWbqfINuEAAsFJAoy0g6uzkCSXtP4\n", - "Oj7q8ABe6ZnO47KudPAns2hUmJPUkv7DsUTmN4GhBWTzeEKCBgrK1GXkzF8EJinA2jkHBvBKzwPA\n", - "73algXbNaZmHF4C1ho1EptQZ6mxgsol4k2EDJe/0xLkQkuxEzyvUE+W9NSelatLzK3jcH6VVuNzx\n", - "NTcnY7aFzIqAoy+/jsTx2QDXnnuW7kgDWA5wLFBkodSFeihYy+7oq8fxki1WankPfjtzXurZM7WJ\n", - "5KZpYGDsrC8p6LOaPP8MNw+ILWGicH0B04ki0vSH88B6u32KR1WmayBBqMfbmmEJk2PUM2suGRDu\n", - "j+VVtuEW4tOYKnA1e45vt3/dMLQWlnwl+Froj1VxQArjZazlPgcmr1vECP10E/gM4FvOnKFrIEGo\n", - "J3bvDEuYpJ4hP9OSXESP5U0JG/Xw+HgVAfZMl5EtG1oLS74SnBSC+3hkDc41DnKsWf/sdIH+aABc\n", - "1Ii3QddAglDPDeKASepJ5+QFi4X743iFTfRQRJ159sz6citlpYMlXwlUDzd7qsXCZVZneKzZ4uOH\n", - "t+hTT+IyxhbMQgsxUqk89Fal0mABPyQ0o6dWwBKdlyqVLyqVhzXheIVN9BRDPbUmfjQrYWHJVwI5\n", - "L4+XR59XuD/fFXAw0PNKbCqHvDQLpZO9z/aAmOT+TFb5qVJz++N4Y8IW0cOetQAXWCu0FpZ8JZD7\n", - "LJ+JE/WE3h803+U+fexjPG/u80m8xhK9zwbSc3qSRXHsfSbdirrheTnejLBF9LDnaDBCKUproepZ\n", - "wTyj8Q1+aOv7+L7ewEhhIocHMosYLvA5toDVGD+KGGtwMJDA6UldFEeSsj8FvK1uqMfxKhsPy50X\n", - "e47VD5FfaS2kHiE4zeDh7TzGpuvMP7X6ImI/zAepF55+la5s59HnqkTxgIOBBE4PTmzl6MUDdDob\n", - "tRx49cTseYW8wuatbx9Z395Xeo959kysfi+V8SCEvD9CkBaZA+1TlznkgJnt/UHslUsUZJLpYNf7\n", - "6l4Uejs9Q9oj4PsSgqFGWNYH3Oxixx0wcyrv/J/LgEkmG5mWiX7gkgPmnZ4hbSd3qmkI7h6wSMNy\n", - "icQS7NrXqkOtMDk3V+1bszvoeu7Qdt6/xL8jQiD/Gw00+X9MzOkyXmS88/nq4GSkrhu6nju0H32p\n", - "94llsXx3bbd3m3H3d0e7vT2T3aXWcz13r/wHdf5/48N/1LIAAAAASUVORK5CYII=\n" - ], - "text/latex": [ - "$$(R \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} + \\dot{\\xi})\\mathbf{i}_{0} + R \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}\\mathbf{j}_{0}$$" - ], - "text/plain": [ - "(R⋅cos(θ)⋅θ̇ + ξ̇) \u001b[94m\u001b[1m0_x\u001b[0;0m\u001b[0;0m + R⋅sin(θ)⋅θ̇ \u001b[94m\u001b[1m0_y\u001b[0;0m\u001b[0;0m" - ] - }, - "execution_count": 54, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Hallamos la velocidad de P en A1, expresada en A0\n", - "# ¡Con esta llamada ya estamos diciendo que C y P son fijos en A2!\n", - "P.v2pt_theory(C, A1, A2).express(A0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Misión cumplida :)**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "---" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "_Hemos hecho un repaso bastante profundo de las posibilidades del paquete `mechanics` de SymPy. Nos hemos dejado algunas cosas en el tintero pero no demasiadas: esta funcionalidad aún se está expandiendo y necesita pulir algunos detalles._\n", - "\n", - "**Referencias**\n", - "\n", - "* Capítulo de **aeromecánica** del libro de Cuerva y otros http://nbviewer.ipython.org/gist/Juanlu001/7711865\n", - "* Estabilidad longitudinal de un Boeing 747 http://nbviewer.ipython.org/github/AlexS12/Mecanica_Vuelo/blob/master/MVII_MatrizSistema.ipynb\n", - "\n", - "_¿Serás tú el siguiente que publique un notebook usando SymPy? ;)_" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Si te ha gustado esta clase:\n", - "\n", - "Tweet\n", - "\n", - "\n", - "---" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "####

¡Síguenos en Twitter!" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "###### Follow @Pybonacci Follow @Alex__S12 Follow @newlawrence " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "##### \"Licencia
Curso AeroPython por Juan Luis Cano Rodriguez y Alejandro Sáez Mollejo se distribuye bajo una Licencia Creative Commons Atribución 4.0 Internacional." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "##### " - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "---\n", - "_Las siguientes celdas contienen configuración del Notebook_\n", - "\n", - "_Para visualizar y utlizar los enlaces a Twitter el notebook debe ejecutarse como [seguro](http://ipython.org/ipython-doc/dev/notebook/security.html)_\n", - "\n", - " File > Trusted Notebook" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/html": [ - "Follow @Pybonacci\n", - "" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "%%html\n", - "Follow @Pybonacci\n", - "" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n", - "El estilo se ha aplicado =)\n", - "\n", - "\n", - "\n" - ], - "text/plain": [ - "" - ] - }, - "execution_count": 1, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Esta celda da el estilo al notebook\n", - "from IPython.core.display import HTML\n", - "css_file = '../styles/aeropython.css'\n", - "HTML(open(css_file, \"r\").read())" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.4.3" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} diff --git a/notebooks_completos/041-SymPy-Mecanica.ipynb b/notebooks_completos/041-SymPy-Mecanica.ipynb new file mode 100644 index 0000000..29a996e --- /dev/null +++ b/notebooks_completos/041-SymPy-Mecanica.ipynb @@ -0,0 +1,1808 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\"AeroPython\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Mecánica con SymPy" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "_Si SymPy te ha parecido hasta ahora un CAS decente e incluso interesante (nada como tener los resultados en $\\LaTeX$ incrustados en el notebook y la sintaxis de Python para hacer cálculo simbólico) entonces espera a ver el paquete `mechanics`. Con él, podremos manipular velocidades y aceleraciones de sólidos expresadas en distintos sistemas de referencia con una facilidad impresionante._\n", + "\n", + "_Tienes disponible la documentación de `mechanics` en http://docs.sympy.org/0.7.5/modules/physics/mechanics/index.html._" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Sistemas de referencia" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "El objeto primordial que vamos a manejar van a ser los sistemas de referencia. Podremos definir relaciones geométricas entre ellos y de esta forma las transformaciones de vectores entre un sistema y otro serán triviales." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "IPython console for SymPy 1.0 (Python 3.5.4-64-bit) (ground types: python)\n", + "\n", + "These commands were executed:\n", + ">>> from __future__ import division\n", + ">>> from sympy import *\n", + ">>> x, y, z, t = symbols('x y z t')\n", + ">>> k, m, n = symbols('k m n', integer=True)\n", + ">>> f, g, h = symbols('f g h', cls=Function)\n", + ">>> init_printing()\n", + "\n", + "Documentation can be found at http://docs.sympy.org/1.0/\n" + ] + } + ], + "source": [ + "from sympy import init_session\n", + "init_session(use_latex=True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Todo lo que necesitamos está en `sympy.physics.mechanics`, incluyendo la clase `ReferenceFrame`. Nada más crear un sistema de referencia podemos acceder a sus versores unitarios: `x`, `y` y `z`.\n", + "\n", + "http://docs.sympy.org/0.7.5/modules/physics/vector/vectors.html" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy.physics.mechanics import ReferenceFrame" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAUBAMAAAB7UtADAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAECLdzUTvdomZVDK7\nq2ayhyTRAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiUlEQVQIHWNgYGBkQABlRzibNcVMAcaRL+BI\ngLGBygVgbGS6xcVSAcrn+sawPwnG/szA8xOmbiMD5xcYW6rnMozN9b8RLq7/D6Rm8X+BfEOG+E8M\nPEA154NSGRj4vzPs+azAwPxtAQMD07UzDee/MzB9QziE574BzGCGu6x/YOxn3xn+Z8E4IBoAX5om\nBf/HSGkAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$\\mathbf{\\hat{a}_x}$$" + ], + "text/plain": [ + "a_x" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A = ReferenceFrame(\"A\")\n", + "A.x" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Y para definir vectores solo tenemos que **multiplicar cada componente por su versor**:" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAWBAMAAABzm9yKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFW3RMUvDQBQH8P+RpKmhxrqIurT2\nCygIdUy+gTe5uHTRwUVxEbeuTirSQUFaVx1S0KkOzeQsLo7xKwhKkyicZxJq3nlvuXs//u/uSIC8\neLGqi+rsTpV8ouSste4DJ6Nz9aysL/kW7BjWN451wbLvAJcAh6ULlv0K2Pe1IQXHXAbtx9GK4lmr\neMCjMzOpa5LUqxNEA3hNXZC404HJsTrUBKm3AOu1HfwFt29/60EOEq/JO6MJ155I/AVsXrzJq2eD\nz5mU/EmWudvvmUkXZhO1RVGHN4QRL3XJQ6u5NwboActHh3sQoendAME1yaHwSuzK542FSLFwsWGI\nEA05SCp388tRPnI7JbFp4+1Ot9mm0jldo1J0m++E3X44lzwTKhpDP69GWej4qml7++NJ6/+Q3R9I\n+wHISmMT/TXw5wAAAABJRU5ErkJggg==\n", + "text/latex": [ + "$$2\\mathbf{\\hat{a}_x} - \\mathbf{\\hat{a}_y}$$" + ], + "text/plain": [ + "2 a_x - a_y" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "2 * A.x - 1 * A.y" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "De ahora en adelante, para trabajar como si nos enfrentáramos a un problema de la escuela, vamos a hacer dos cosas:\n", + "\n", + "* Definir un sistema inercial $1$ del que partir, para así poder referir todos los demás sistemas a él.\n", + "* Que los versores de ese sistema sean $i, j, k$." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAATBAMAAADMlCYHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu91EzRCrMiKZVHaJ\n72YDRvcCAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVQoFWNgEDJgQAG9qihc9v+/wXzm/w4o\n4gzz/6HyayHqWP8noIq//47KZ4GoYyiGCq+G0lxQd3BtgAjA1MF0w9TB+Gjq+GH2QtUx//8DUYmk\nbj1QDYd/AkQcZl4TprreVKCS+WjqGDHUsd8BmURYnf0DuLojLi66Li5uID7YPB4XF6crLi4JQD7L\nl/8girB5n96DQ56gvb+4/zcQY94vBnkF4tT1/2Rg4P//H6gYCGDhhxYu7P//O5z//wOiBqGOtQBN\nHUIBlBUBpvm/8/6CCLA2QCWwUvyf6xWwSqAJMjtpBqAJgbkA41Varlo3R+MAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$\\mathbf{i} + \\mathbf{j} + \\mathbf{k}$$" + ], + "text/plain": [ + "1_x + 1_y + 1_z" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A = ReferenceFrame(\"1\", latexs=['\\mathbf{i}', '\\mathbf{j}', '\\mathbf{k}'])\n", + "A.x + A.y + A.z" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Y para no tener que hacerlo siempre, un pequeño truco de magia:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "# Definimos nuestra propia clase para que los versores sean IJK\n", + "# aeropython: preserve\n", + "class IJKReferenceFrame(ReferenceFrame):\n", + " def __init__(self, name):\n", + " super().__init__(name, latexs=['\\mathbf{%s}_{%s}' % (idx, name) for idx in (\"i\", \"j\", \"k\")])\n", + " self.i = self.x\n", + " self.j = self.y\n", + " self.k = self.z" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGcAAAATBAMAAAB/3A4dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu91EzRCrMiKZVO92\niWZOMaHgAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABK0lEQVQoFWNgEDJgwAAcmwIwxNj//4GLMf93\ngLPhDM7/D+BsOKMaoYn1fwJcGM7g+18AZ8MZLAhNDMVwUSBjDZSzGIsgsiZkabgmbIJImvhRnAe1\nyf7/BSRtUEGgpvVQxRz+CZjyHPrYNXWnQtXOB2oqgemD+Un+AgO7A5ogyx/2vUAhsDhQU2USmjyD\n/AX2aQpogix/7B8A9YDFQTYtAstfcXHRdnFxA7GBNnEpoAmyfAH7CCyO0ARUhOQ8qCaEIMun9/+A\nPBI1/eL530C6JgZ5BTI09f+EaOL//x8WEAjnIwUEXJD9/3+H+/9/wAIIQ1M4Nk1A3WAAD6DVMJEI\nMON/wf4FDFwbUAVhPJh4p34OTAiiyfwTA8ekbwdQBKEcXOIMJkpTsamHiwEA3YRstqP1a+gAAAAA\nSUVORK5CYII=\n", + "text/latex": [ + "$$\\mathbf{i}_{1} + \\mathbf{j}_{1} + \\mathbf{k}_{1}$$" + ], + "text/plain": [ + "1_x + 1_y + 1_z" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A = IJKReferenceFrame(\"1\")\n", + "A.i + A.j + A.k" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Álgebra vectorial" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Nuestros vectores funcionan también con símbolos, y podemos realizar las operaciones de producto escalar y producto vectorial con ellos." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "R, V = symbols('R, V', positive=True)\n", + "r1 = R * (A.x + A.y + A.z)\n", + "v1 = V * (A.x - 2 * A.z)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAAATBAMAAABvkIKEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "text/latex": [ + "$$R\\mathbf{i}_{1} + R\\mathbf{j}_{1} + R\\mathbf{k}_{1}$$" + ], + "text/plain": [ + "R 1_x + R 1_y + R 1_z" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "r1" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAASBAMAAABFi4Y1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4RVtAAAAAElFTkSuQmCC\n", + "text/latex": [ + "$$V\\mathbf{i}_{1} - 2 V\\mathbf{k}_{1}$$" + ], + "text/plain": [ + "V 1_x - 2⋅V 1_z" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "v1" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy.physics.mechanics import dot, cross" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAOBAMAAABN+VgMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\niUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyUlEQVQYGWNggAHW/G/p5QEMk/9rMzA4/zoA\nEwbS/AUMHN8YGD8BmdwJSOIM/hMYGPIdGFYBxdiQxRnmOzAw2Dcw6AMFp6BIVAGN+A7U1MDAcQFF\n4hsDw64EBob7D9BMYvk2s/4IUGX8BDSTOD8ysCoCJd4fgJgkZAwCKgwM7AsYGP4CJeQLUN3EwJ8A\n8gbQOwZxQBIJ+AcwcP8B8pl/HEASBTKB3uD9ySDAwP5dAFUC6DHWnywbGHj+oYof+p/KwHB8JlA2\nASwBADazL5flHdcjAAAAAElFTkSuQmCC\n", + "text/latex": [ + "$$- R V$$" + ], + "text/plain": [ + "-R⋅V" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "r1.dot(v1)\n", + "dot(r1, v1)\n", + "r1 & v1" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAAATBAMAAACO+5fVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "text/latex": [ + "$$- 2 R V\\mathbf{i}_{1} + 3 R V\\mathbf{j}_{1} - R V\\mathbf{k}_{1}$$" + ], + "text/plain": [ + "- 2⋅R⋅V 1_x + 3⋅R⋅V 1_y - R⋅V 1_z" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "r1.cross(v1)\n", + "cross(r1, v1)\n", + "r1 ^ v1" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Podemos hallar también la norma de los vectores con su método `magnitude` e incluso normalizarlos con `normalize`:" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAVBAMAAADiLT7tAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n82gOg8SBsAAAAABJRU5ErkJggg==\n", + "text/latex": [ + "$$\\sqrt{14} R V$$" + ], + "text/plain": [ + "√14⋅R⋅V" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(r1 ^ v1).magnitude()" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQIAAAAwBAMAAAAIpGlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRobahpHkRDEXnC23kMRfgm+bwUaZb2MAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$- \\frac{\\sqrt{14}}{7}\\mathbf{i}_{1} + \\frac{3 \\sqrt{14}}{14}\\mathbf{j}_{1} - \\frac{\\sqrt{14}}{14}\\mathbf{k}_{1}$$" + ], + "text/plain": [ + "- √14 \n", + "─────\n", + " 7 1_x + 3⋅√14\n", + "─────\n", + " 14 1_y - √14 \n", + "─────\n", + " 14 1_z" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(r1 ^ v1).normalize()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "##### Ejercicio\n", + "\n", + "Usando directamente la fórmula para la derivada en ejes móviles:\n", + "\n", + "$$\\left(\\frac{\\operatorname{d}\\!\\mathbf{a}}{\\operatorname{d}\\!t}\\right)_1 = \\left(\\frac{\\operatorname{d}\\!\\mathbf{a}}{\\operatorname{d}\\!t}\\right)_0 + \\mathbf{\\omega}_{01}\\! \\times \\mathbf{a}$$\n", + "\n", + "Calcula la derivada del vector de posición $R \\mathbf{i}_0$, siendo $A_0$ un sistema de referencia que gira respecto al inercial con velocidad angular $\\mathbf{\\omega}_{01}=\\Omega \\mathbf{k}_0$. **¿Cuál es el módulo de la derivada?**" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "R, Omega = symbols('R, Omega', positive=True)\n", + "A0 = IJKReferenceFrame('0')" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAASBAMAAABRHhjUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqu7IjJ23e9UmRDN\niUTiMbt4AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVQYGU2OMW7CQBBFnyDGBANrUSfKioIa\nhES4BWlpyEp0aXKDKMoJEBeAHiF8BG5gKlp8BAKYIhTLLmQxU828+TP/g6vXxHU1XW+3qOqlA+yh\nJwMtHfCOkEsZuJlCCv7hNkL+GyZNkZ0IyeMPD7F0ovfG8O8NFpLCemXhZ8RTdAHPjC3ogjAbo/hC\nmA07G8OCIKU4Au/XmEwt8A74IZcYGxVYcMJXUN7Ci+pfFUVFKd6FlDpKaG1+iIT7+nfJ0Mc1RwYq\n8xlnXb01JCaHWEQAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$R\\mathbf{i}_{0}$$" + ], + "text/plain": [ + "R 0_x" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "a = R * A0.i\n", + "a" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAASBAMAAADBDWEbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmaJu6sQ\n791S4TDfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVQYGVXOMU4CQRTG8f+wrMuKIVvZGTUe\nQCsTKrkBVNpyBC9glniBNYRQ2HgEPQAFN9BoZWMgNLawQlYljt9IAuMrZl5+efPNA8zRIUQ2Z13l\n82T3Aq48yjpwdkC4oWiu+XDq09anKM59CmeiypejY/ugHrYLHcG3o73BnxAsdFcLUfSqrl6HnTc1\ntSdRs6OEm5IeZwmkWmLukmoNllrgBG51/WRKfbzkGWNaMDaEs8AmpAnvxAww7VjE/Yh0yITqdY9S\n99TR/sJNvRDbgorNI2tbd3aqrA9t4NfqR18I2m6v/9XvD38B5S9IY52D59QAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$\\Omega\\mathbf{k}_{0}$$" + ], + "text/plain": [ + "Ω 0_z" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "omega01 = Omega * A0.k\n", + "omega01" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmaJu6sQ\n791S4TDfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABNElEQVQYGVWQsUvDUBDGv5dok7TSZnKO6CoU\nCkKnuoijTroIdhFBHPIfNPgP9CHSwclZkHQUEZrNUdG9Ii5OFo1KbYrPL+8JkoPcfe939+7uBYBY\nXMC/iWFiDjNb/vw2pTvIrs4jlFRkeJexFVDXziAyuCrU3PlkmH2j69SBQYgljVH6ZvRSuhYLR76h\nLH2ncid0F4A1wYaSOlUeM9gZHb/lPsTIcPuLqMKkkzXifepYotkE5obU1TvemcLd0dyVVh/o+lwl\nAMrM5TvEsrqKKQfXgVOea+ydj4nlbYh7CLEJPAneiWDls2LZ8fECD9cQbU+vz2ckOU/wjMrRCazj\nFWCXz0idwNQ/wFNj/pMUe+oSOGj89f9gw4LdsN7sU8BQ4WNgt/P9i6bWf4BeLylSYO31UKNfnF5f\nvJfxI8sAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$\\Omega R\\mathbf{j}_{0}$$" + ], + "text/plain": [ + "Ω⋅R 0_y" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "da = omega01 ^ a\n", + "da" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAPBAMAAADuVgmRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmaJu6sQ\n791S4TDfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0klEQVQIHU2OsUrDUBSGvxsbkzRiMnWOOBcC\nga55BSddBF1E6NQ3aPAFAuLk5FwQM5ZSSB+hT1DoE4gGJRoh/snUC/c/H/89554fMOdnHJ7BZTi6\nkuGWzWqRCXJJGgmCF0wDzpfY/pDMYyhnHP+IvUqSzuA9xP4Uu7+SV7BUh7X4SLPojgvxt9iX6zTJ\n2734ZCc53erlD/daTB5qQaRZmd0C7BieVYOij4ExF7A36sqwuv881phbr4+hOBv8hyesxwncKE7l\nRHhtjdtW3LVLmCb8A/w5NMBDzH8OAAAAAElFTkSuQmCC\n", + "text/latex": [ + "$$\\Omega R$$" + ], + "text/plain": [ + "Ω⋅R" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "da.magnitude()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "
Si no especificaste `positive=True` vas a ver algo como $\\sqrt{\\Omega^2 R^2}$. Debería haber una forma de simplificar esta expresión _a posteriori_, pero de momento no funciona del todo bien. Preparando este notebook nos hemos dado cuenta y ya les hemos avisado :) https://github.com/sympy/sympy/issues/8326\n", + "
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Movimiento relativo" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "¿A quién no le gusta multiplicar matrices de rotación? Para esa minoría que lo detesta, existe SymPy. Para ello debemos especificar la orientación de nuestros sistemas de referencia usando el método `orient`, y recuperaremos la matriz de cosenos directores usando el método `dcm`." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "A1 = IJKReferenceFrame(\"1\")\n", + "A0 = IJKReferenceFrame(\"0\")" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMwAAABLCAMAAAALSZigAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAABN9JREFUaAXtW9uW\nmzAMNGBoyzXU//+vteQrrJcgWdk9bcNDjnFgrJFk2UyIagwerfqLj4fjoFRjOm2P/i/mojZg0Bog\n05B5zFu8ZYbWhp+xj92owt14ZOY9mTthsyWxaRadALIWGXfopm0KsWCSWTPTHZlhzWx62myMu+t8\nIRl3tZNjeHhreGS2LlkxeB9PZV+nC2+0yLjbAqiTdySPTHAFII0+yAPiQg//IOPuWIJ7M+CYLDLz\nI7M35ss6Zr1PmnMT8jy/kI5rkMxoXCUuk7HTaurggmHSegIjG91vOkwLHbOsGedu7l3KdrE3N/DY\n9jDzDlY069I1tqDG/CTjDgbHHI2DKJKZF0ukWWbVYNx3rQasXiEge6jLljDUUjdtt8D1aH9+lmBW\ndOnDQqvZhGpCxp1dHQnlpEhmBbN6u/o8MIfsOjQukJXBhQ8XVQWDwwIzucwNXHPzj+0E45J9x3lm\ngm/IuLOLzBUZHzxYTZ3Vlv/D7Do4UC1udmxgPdDtkEZjl6xnR4TxZDCYPkkUHfdGmoUUtMFxVpvd\nTp7VODdYez2ZHRIWyLicuUMmwngymL1nMgRcVwD6iwLwMTLdDFm2eW7Kp8Ni0wOmzOywxudplmCK\nZOi4O3pjuyrNOGfsrFGLLxZjj9Ol9cntJ+pqO2HKoCdt83kBSDBFMnRct2h2fuRiAZjB52qy1WyB\n5WCdVI8FYPezZnJFGEJhs8xzUR1WC3v950eCcS51nzHNGLgQzAGttKMWyaih7bTlYjPILjitnTjj\ntOkNFxywNCRU02277sLyt4YGXFI+AkzTGtMO+Nn0u1mwGrJwYSFsw8BlMmVTUm/cduQ79udTJgF8\n0qrE5ZGJG8LsMUZ0o8nD5ZFRcfZEF9MeAeJtp0YdLpNMeIjy5c2aFMw4WUc8rcNlksH1JTc0y4u8\nm9zOJiHeS8LlkiFb+RU3vMl8hZc5Y7wjw/HaV9xTHZms/ODOjVR+Cgxr8O6SEVPtCvbnXWGhgT63\ncb1WF1kiYHgyzQeGNlm1OwOczql4AiJgtCBu0myPhBpIxZMQASOZuM21PRJqIBXPPeNRRUAp1S66\nodggq4B3RMA0krRqJ6su3lBnEpVM/BNR7aTVxTsiYGIjrNpJq4sXIuCwPuIRHuhFVbuPGlalulhI\nsx8/f6VYnFqiqp28ungUAX//BBU2yBsnJladieJfUegiqnYfI1OrLt4QARMnYdVOXF28IQJmZKL4\nJ6LayauLN0TAyEZatRNXFwVEwEg2bj/ynXuFGliHd/cRIJp/bMSNYfYYU6MG1uFVkolyWRLN69TA\nIL+x8GrJhIcpKTWwCq+WjLgamE0+zOgsf48ZXjirJlPA/LauN5lvc/2Tgd+ReeKgb/v6f47MQXOT\njIAIMDEyB81Nkgwf2P0cj7bQyBw1N0EyXGC7C2/96xnWGhqZo+YmSKYCWHPJHDU3QTIVwFwyJzFE\njkwNMJfMSXOTI1MDzCeDbwB99vMGn9xJzCMBccnUZMOlgTXAXDLqqLld2kf7sgKYTeaoudHsvby6\nAphN5qi5XZpH+7ICmE0G386ML97R7L2++iDmXV96+pZP5qC5nVCrTrnAXbuYNbw7StvOVNn7+pvf\nZF7vY94I78jw/Pb6u/7ByPxDfzod4A+bWrt/ZLw+GV4yAv7pVGv1B5tiTdWO22oCAAAAAElFTkSu\nQmCC\n", + "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )} & \\sin{\\left (\\phi \\right )} & 0\\\\- \\sin{\\left (\\phi \\right )} & \\cos{\\left (\\phi \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡cos(φ) sin(φ) 0⎤\n", + "⎢ ⎥\n", + "⎢-sin(φ) cos(φ) 0⎥\n", + "⎢ ⎥\n", + "⎣ 0 0 1⎦" + ] + }, + "execution_count": 22, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "phi = symbols('phi')\n", + "A0.orient(A1, 'Axis', [phi, A1.z]) # Rotación phi alrededor del eje A1.z\n", + "A0.dcm(A1) # \"Direct Cosine Matrix\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Usando el argumento `Axis` hemos especificado que rotamos el sistema un ángulo especificado alrededor de un eje. Otros métodos son:\n", + "\n", + "* `Body`: se especifican los tres ángulos de Euler.\n", + "* `Space`: igual que `Body`, pero las rotaciones se aplican en orden inverso.\n", + "* `Quaternion`: utilizando cuaternios, rotación alrededor de un vector unitario $\\lambda$ una cantidad $\\theta$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "
¿Qué es lo bueno de usar uno de estos métodos? ¡Que **siempre** tenemos la transformación bien definida! Es imposible meter \"a capón\" una matriz de rotación que sea incorrecta o absurda.
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Diferente sistema de referencia\n", + "\n", + "Para expresar un vector en otro sistema de referencia, no hay más que usar los métodos `express` o `to_matrix`:" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAUBAMAAAA0IxGWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "text/latex": [ + "$$\\operatorname{cos}\\left(\\phi\\right)\\mathbf{i}_{1} + \\operatorname{sin}\\left(\\phi\\right)\\mathbf{j}_{1}$$" + ], + "text/plain": [ + "cos(φ) 1_x + sin(φ) 1_y" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A0.x.express(A1)" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAABLCAMAAADtTJGhAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )}\\\\\\sin{\\left (\\phi \\right )}\\\\0\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡cos(φ)⎤\n", + "⎢ ⎥\n", + "⎢sin(φ)⎥\n", + "⎢ ⎥\n", + "⎣ 0 ⎦" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A0.x.to_matrix(A1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Símbolos dinámicos\n", + "\n", + "Si queremos especificar que un símbolo puede variar con el tiempo, hay que usar la función `dynamicsymbols`:" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy.physics.mechanics import dynamicsymbols" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAUBAMAAAD1iJl/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXaZRCLdEO9Uu81m\nqzIdlvb2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVQYGWNggAHWAyDWURgXREeBObwBSGJb\nGRjYUxnYNyCEeAwYGDg/MDAIIYTYgDqYgUoqEEJhQCabAAMDtwOQIaQMIpUZGHzXr1VgYDZgYAhz\nYG9lncCwBig8H4h5gJobGRgWMxUwZAK51kDMmcDA/B3IZGRgaAFyE4GY7wMDxwIGBglxsBD7F4hQ\nPdAW+QkMII0sHxgKGFgSGPgPAIWAskDjGTdwPgAZz3yBgceer4BBGOjSC0VAAqhL8MwBZqAv3BgY\nWNcAdXA/AGqBAJCHQADkCygAeRsEkLzNsBQsghw40CDkguoHy8MDGgDDjSpJfBFnNAAAAABJRU5E\nrkJggg==\n", + "text/latex": [ + "$$\\alpha{\\left (t \\right )}$$" + ], + "text/plain": [ + "α(t)" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "alpha = dynamicsymbols('alpha')\n", + "alpha" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Y pedir su derivada con el método `diff`:" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAArBAMAAAAu4Q2VAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWAYkAAAAAElFTkSuQmCC\n", + "text/latex": [ + "$$\\frac{d}{d t} \\alpha{\\left (t \\right )}$$" + ], + "text/plain": [ + "d \n", + "──(α(t))\n", + "dt " + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "alpha.diff()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "##### Ejercicio\n", + "\n", + "![Sistemas de ejes asociados a la pala](../images/ejes_pala.jpg)\n", + "\n", + "(Sacado de Cuerva et al. \"Teoría de los Helicópteros\")\n", + "\n", + "**Obtener la matriz de rotación de la pala $B$ respecto a los ejes $A1$.**" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [], + "source": [ + "A = IJKReferenceFrame(\"A\")" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQAAAABLCAMAAACoRKlPAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nK/4DNGVhyUtpMY8AAAAASUVORK5CYII=\n", + "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\psi{\\left (t \\right )} \\right )} & \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡cos(ψ(t)) sin(ψ(t)) 0⎤\n", + "⎢ ⎥\n", + "⎢-sin(ψ(t)) cos(ψ(t)) 0⎥\n", + "⎢ ⎥\n", + "⎣ 0 0 1⎦" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A1 = IJKReferenceFrame(\"A1\")\n", + "psi = dynamicsymbols('psi')\n", + "A1.orient(A, 'Axis', [psi, A.z])\n", + "A1.dcm(A) # T_{A1A}" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\sin{\\left (\\beta{\\left (t \\right )} \\right )}\\\\0 & 1 & 0\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡cos(β(t)) 0 sin(β(t))⎤\n", + "⎢ ⎥\n", + "⎢ 0 1 0 ⎥\n", + "⎢ ⎥\n", + "⎣-sin(β(t)) 0 cos(β(t))⎦" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A2 = IJKReferenceFrame(\"A2\")\n", + "beta = dynamicsymbols('beta')\n", + "A2.orient(A1, 'Axis', [beta, -A1.y])\n", + "A2.dcm(A1) # T_{A2A1}" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡cos(β(t))⋅cos(ζ(t)) sin(ζ(t)) sin(β(t))⋅cos(ζ(t)) ⎤\n", + "⎢ ⎥\n", + "⎢-sin(ζ(t))⋅cos(β(t)) cos(ζ(t)) -sin(β(t))⋅sin(ζ(t))⎥\n", + "⎢ ⎥\n", + "⎣ -sin(β(t)) 0 cos(β(t)) ⎦" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A3 = IJKReferenceFrame(\"A3\")\n", + "zeta = dynamicsymbols('zeta')\n", + "A3.orient(A2, 'Axis', [zeta, A2.z])\n", + "A3.dcm(A1) # T_{A3A1}" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\0 & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡1 0 0 ⎤\n", + "⎢ ⎥\n", + "⎢0 cos(θ(t)) sin(θ(t))⎥\n", + "⎢ ⎥\n", + "⎣0 -sin(θ(t)) cos(θ(t))⎦" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "B = IJKReferenceFrame(\"B\")\n", + "theta = dynamicsymbols('theta')\n", + "B.orient(A3, 'Axis', [theta, A3.x])\n", + "B.dcm(A3) # T_{BA3}" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & 0\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\\\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ cos(ζ(t)) sin(ζ(t)) 0 ⎤\n", + "⎢ ⎥\n", + "⎢-sin(ζ(t))⋅cos(θ(t)) cos(θ(t))⋅cos(ζ(t)) sin(θ(t))⎥\n", + "⎢ ⎥\n", + "⎣sin(θ(t))⋅sin(ζ(t)) -sin(θ(t))⋅cos(ζ(t)) cos(θ(t))⎦" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "B.dcm(A2)" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ cos(β(t))⋅cos(ζ(t)) sin(ζ(t)) \n", + "⎢ \n", + "⎢-sin(β(t))⋅sin(θ(t)) - sin(ζ(t))⋅cos(β(t))⋅cos(θ(t)) cos(θ(t))⋅cos(ζ(t)) -\n", + "⎢ \n", + "⎣-sin(β(t))⋅cos(θ(t)) + sin(θ(t))⋅sin(ζ(t))⋅cos(β(t)) -sin(θ(t))⋅cos(ζ(t)) s\n", + "\n", + " sin(β(t))⋅cos(ζ(t)) ⎤\n", + " ⎥\n", + "sin(β(t))⋅sin(ζ(t))⋅cos(θ(t)) + sin(θ(t))⋅cos(β(t))⎥\n", + " ⎥\n", + "in(β(t))⋅sin(θ(t))⋅sin(ζ(t)) + cos(β(t))⋅cos(θ(t)) ⎦" + ] + }, + "execution_count": 34, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "B.dcm(A1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### Velocidad angular\n", + "\n", + "También podemos hallar la velocidad angular de un sistema respecto a otro usando el método `ang_vel_in`:" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2}$$" + ], + "text/plain": [ + "d \n", + "──(θ(t))\n", + "dt a3_x + d \n", + "──(ζ(t))\n", + "dt a2_z" + ] + }, + "execution_count": 35, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "B.ang_vel_in(A2)" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2} - \\dot{\\beta}\\mathbf{j}_{A1} + \\dot{\\psi}\\mathbf{k}_{A}$$" + ], + "text/plain": [ + "d \n", + "──(θ(t))\n", + "dt a3_x + d \n", + "──(ζ(t))\n", + "dt a2_z + d \n", + "-──(β(t))\n", + " dt a1_y + d \n", + "──(ψ(t))\n", + "dt a_z" + ] + }, + "execution_count": 36, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "B.ang_vel_in(A)" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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zZ1P+/vjnpHXTX08Jz3nAdv/x4Udm6Kpptq6vJx+74coBbYnX+9bdA3nNI9/W+BiTViku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\nRDCzk/8G1kC1zVyaqU8AAAAASUVORK5CYII=\n", + "text/latex": [ + "$$(\\left(- \\operatorname{sin}\\left(\\psi\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\psi\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\psi\\right) \\dot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\psi\\right) \\dot{\\beta})\\mathbf{i}_{A} + (\\left(\\operatorname{sin}\\left(\\psi\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\psi\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\psi\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\psi\\right) \\dot{\\beta})\\mathbf{j}_{A} + (\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\theta} + \\operatorname{cos}\\left(\\beta\\right) \\dot{\\zeta\\left(t\\right)} + \\dot{\\psi})\\mathbf{k}_{A}$$" + ], + "text/plain": [ + "⎛ d d d ⎞\n", + "⎜(-sin(ψ(t))⋅sin(ζ(t)) + cos(β(t))⋅cos(ψ(t))⋅cos(ζ(t)))⋅──(θ(t)) - sin(β(t))⋅cos(ψ(t))⋅──(ζ(t)) + sin(ψ(t))⋅──(β(t))⎟\n", + "⎝ dt dt dt ⎠ a_x + ⎛ d d d ⎞\n", + "⎜(sin(ψ(t))⋅cos(β(t))⋅cos(ζ(t)) + sin(ζ(t))⋅cos(ψ(t)))⋅──(θ(t)) - sin(β(t))⋅sin(ψ(t))⋅──(ζ(t)) - cos(ψ(t))⋅──(β(t))⎟\n", + "⎝ dt dt dt ⎠ a_y + ⎛ d d d ⎞\n", + "⎜sin(β(t))⋅cos(ζ(t))⋅──(θ(t)) + cos(β(t))⋅──(ζ(t)) + ──(ψ(t))⎟\n", + "⎝ dt dt dt ⎠ a_z" + ] + }, + "execution_count": 37, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "B.ang_vel_in(A).express(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Derivada en ejes móviles\n", + "\n", + "Hacer una derivada con la fórmula lo hace cualquiera, pero SymPy puede encargarse automáticamente." + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAASBAMAAAC3N9OQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu91EzRCrMiKZZol2\nVO8rb2IPAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgEDJgQADm/w4IDgPr/wQkHkMx\nMgeVzY+iksMfRd98MI+xgYGhBKgLwltvwFCZBOOxzl/AwHAJxuMOmoDEq2EUQOIlcCogeFwXeD6A\nefz//zMwBM2c+BUiBzSIYQYD+3c4j3cBA8MvBoZ7IAkGBnkBhsZPCzr1cwCP8yYDTbxFiQAAAABJ\nRU5ErkJggg==\n", + "text/latex": [ + "$$\\mathbf{i}_{A1}$$" + ], + "text/plain": [ + "a1_x" + ] + }, + "execution_count": 38, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "v1 = A1.x\n", + "v1" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAAAZBAMAAAD9ML7rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "text/latex": [ + "$$(\\operatorname{sin}^{2}\\left(\\psi\\right) \\dot{\\psi} + \\operatorname{cos}^{2}\\left(\\psi\\right) \\dot{\\psi})\\mathbf{j}_{A1}$$" + ], + "text/plain": [ + "⎛ 2 d 2 d ⎞\n", + "⎜sin (ψ(t))⋅──(ψ(t)) + cos (ψ(t))⋅──(ψ(t))⎟\n", + "⎝ dt dt ⎠ a1_y" + ] + }, + "execution_count": 39, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "#v1.diff(dynamicsymbols._t, A2)\n", + "dv1 = v1.diff(symbols('t'), A)\n", + "dv1" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/latex": [ + "$$\\left[\\begin{matrix}0\\\\\\sin^{2}{\\left (\\psi{\\left (t \\right )} \\right )} \\frac{d}{d t} \\psi{\\left (t \\right )} + \\cos^{2}{\\left (\\psi{\\left (t \\right )} \\right )} \\frac{d}{d t} \\psi{\\left (t \\right )}\\\\0\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡ 0 ⎤\n", + "⎢ ⎥\n", + "⎢ 2 d 2 d ⎥\n", + "⎢sin (ψ(t))⋅──(ψ(t)) + cos (ψ(t))⋅──(ψ(t))⎥\n", + "⎢ dt dt ⎥\n", + "⎢ ⎥\n", + "⎣ 0 ⎦" + ] + }, + "execution_count": 40, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "dv1.to_matrix(A1)" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAArBAMAAADf6lYYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "text/latex": [ + "$$\\frac{d}{d t} \\psi{\\left (t \\right )}$$" + ], + "text/plain": [ + "d \n", + "──(ψ(t))\n", + "dt " + ] + }, + "execution_count": 41, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(dv1 & A1.j).simplify()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Puntos, velocidades y la rueda que no desliza" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "El último paso que nos queda para completar la cinemática es la posibilidad de definir puntos en sólidos y aplicar su campo de velocidades. SymPy también permite esto, y para ello no tenemos más que importar la clase `Point`." + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy.physics.mechanics import Point" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "metadata": {}, + "outputs": [], + "source": [ + "O = Point(\"O\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Para trabajar como lo haríamos en la escuela, vamos a especificar que $O$ es el origen de $A$, y para eso vamos a imponer que su velocidad es cero con el método `set_vel`:" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": {}, + "outputs": [], + "source": [ + "O.set_vel(A, 0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Para definir nuevos puntos, podemos utilizar el método `locate_new`:" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": {}, + "outputs": [], + "source": [ + "e_b = symbols('e_b')\n", + "E_b = O.locatenew('E_b', e_b * A1.x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Y para obtener vectores de un punto a otro, el método `pos_from`:" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAACwAAAASBAMAAADS2iHvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVQYGWNggIPnCXAmEoPt/wUkHpzJ+H8D\nnI3MWIbMIcTmRzWEUdkkAKSF9T6K2ekBrApsd4Hi/mBhPqCatUBFnQwiCxhU4MLnExiWbGdg4O4x\nPsjAcBMmzOh/gIFhOgMDP1jrV5gwk7gDRFgBKML22VoAYvZivgdgYQ6gsAFzIbMDRHgDSwFYmPkC\ng0gAnwHPBbAwzwTmD2BhBhObAwwcCSwb+P//Z2AQd/H7CBEGGs3AL8AxAUQzuDJw/UIIswucA4ty\nHmBg+MPAMAPMYWDg2qQAZr1/wBD3/UBI/WYAKbY7VOKAM4EAAAAASUVORK5CYII=\n", + "text/latex": [ + "$$e_{b}\\mathbf{i}_{A1}$$" + ], + "text/plain": [ + "e_b a1_x" + ] + }, + "execution_count": 46, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "E_b.pos_from(O)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "
La notación de este paquete está influenciada por el libro Kane, T. R. & Levinson, D. A. \"Dynamics, Theory and Applications\". Es ligeramente distinto a como estudiamos nosotros en la escuela, pero ¡están abiertos a que les hagamos cualquier tipo de sugerencia! https://github.com/sympy/sympy/issues/2584#issuecomment-31552654
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Por último, el **campo de velocidades de un sólido rígido** se formula usando el método `v2pt_theory`.\n", + "\n", + "$$v^P_A = v^O_A + \\omega_{A_1 A} \\times \\mathbf{OP}$$\n", + "\n", + "Este método pertenece *al punto del cual queremos conocer la velocidad* y recibe tres parámetros:\n", + "\n", + "* `O`, punto de velocidad conocida respecto a A\n", + "* `A`, sistema de referencia donde queremos calcular la velocidad\n", + "* `A1`, sistema de referencia donde están fijos ambos puntos (_sistema de arrastre_)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Por tanto, para hallar la velocidad del punto que acabamos de crear:" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAYBAMAAABZ8fMmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "text/latex": [ + "$$e_{b} \\dot{\\psi}\\mathbf{j}_{A1}$$" + ], + "text/plain": [ + " d \n", + "e_b⋅──(ψ(t))\n", + " dt a1_y" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "E_b.v2pt_theory(O, A, A1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "##### Ejercicio\n", + "\n", + "![Disco que rueda](../images/disco.png)\n", + "\n", + "(Apuntes de Óscar López Rebollal)\n", + "\n", + "**¡Halla la velocidad y la aceleración de $P$!**" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": {}, + "outputs": [], + "source": [ + "# Creamos nuestros sistemas de referencia\n", + "A1 = IJKReferenceFrame('1')\n", + "A0 = IJKReferenceFrame('0')\n", + "A2 = IJKReferenceFrame('2')" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAUBAMAAACezBVvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny//EX5RzbwtR0hgsAAAAAElFTkSuQmCC\n", + "text/latex": [ + "$$\\left ( \\xi{\\left (t \\right )}, \\quad \\theta{\\left (t \\right )}\\right )$$" + ], + "text/plain": [ + "(ξ(t), θ(t))" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Creamos los símbolos dinámicos necesarios\n", + "xi, theta = dynamicsymbols('xi, theta')\n", + "xi, theta" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "metadata": {}, + "outputs": [], + "source": [ + "# Orientamos los sistemas de referencia\n", + "A0.orient(A1, 'Axis', [0, A1.k]) # A0 no gira respecto a A1\n", + "A2.orient(A0, 'Axis', [theta, A0.k])" + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQuM/AAAAAElFTkSuQmCC\n", + "text/latex": [ + "$$\\left[\\begin{matrix}\\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" + ], + "text/plain": [ + "⎡cos(θ(t)) sin(θ(t)) 0⎤\n", + "⎢ ⎥\n", + "⎢-sin(θ(t)) cos(θ(t)) 0⎥\n", + "⎢ ⎥\n", + "⎣ 0 0 1⎦" + ] + }, + "execution_count": 51, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "A2.dcm(A1)" + ] + }, + { + "cell_type": "code", + "execution_count": 52, + "metadata": {}, + "outputs": [], + "source": [ + "# Creamos el punto C, centro del disco, y especificamos su velocidad\n", + "# respecto a A1\n", + "C = Point('C')\n", + "C.set_vel(A1, xi.diff() * A1.x)" + ] + }, + { + "cell_type": "code", + "execution_count": 53, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\niUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA50lEQVQYGWNgQAMciwVQRVjzv6WXBzDw/A9A\nFWfgL2Dg+MbA+t8BTdx/AgNDvgPDPjRhhvlAhfYN6KIMDFUMDNzfGd7/v4Am9Y2BYVcCA4c9mjjL\nt5n1R4BK6y8w3KkRQOjh/MjAqggW521gU2AQMgYBFQYG9gUMDH/B4swNLD8Q6vmBZgOtAJrD/IDj\nH0LcP4CB+w9YHOiujwhxoPN5fzIIgOxlYJ6AENdnYGD9ybIBLH4HIXzofyoDw/GZYHeyHkCIQ1ki\nIPVvGZ6gS/x3WL+BRXt3IYb4k88MXP//f0AXf2R8ECwEAJWaRfWXrUh4AAAAAElFTkSuQmCC\n", + "text/latex": [ + "$$- R\\mathbf{j}_{2}$$" + ], + "text/plain": [ + "- R 2_y" + ] + }, + "execution_count": 53, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Localizamos el punto P, punto fijo del disco, respecto a C, en\n", + "# el sistema A2 (que gira solidariamente con el disco)\n", + "R = symbols('R')\n", + "P = C.locatenew('P', -R * A2.j)\n", + "P.pos_from(C)" + ] + }, + { + "cell_type": "code", + "execution_count": 54, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAR4AAAAZBAMAAAAVlVzpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRImr\nMhBQc+7aAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEtElEQVRIDe1WXWibZRR+vvw1S5sfpiheyL6J\nuxCERb0oopgIQRCFxasqonTiv7IFdfSiwqKggjet4twYowso4o00inX4A83dEArN1VxFaBQGis50\n6uboqvE5533fL1/SpJt3XvhCv+855znnOed7/1LgX41Eoz98s6c/4grs568gZmBIu9jv3uzpj7i8\nnfnj8jH/R/xHZ8Dzt2psO8kffOy0MYR941coGX22z7+l6flA5gaTyUCB3rstIo6r+HdzZ+m2W9Xq\ne3h5IHXXaxjLKSGwb7wCJecHbsmw7kqlmyk13/ux7GQFxju+4e+W1+/Ajoaxe54x9rHSyrYyRXUL\n7OGRvnNayfZGr99aId3VejeCNdNz+BtGVmGk01Q+UuYr/Se/8q9ufIB2ET2GbBUn1KVQULKiNrA4\nMQUh0zXrkNcbDg/RlZoJHxc108E3TVIix/cIe0mccyqh914gMousj5/EaaAg10+EHwJDCrAj6GeI\nrtS8CfjNZhpoc6+Rd3QWmMlbT+jlTbLPMlariNXoNlB410+MiRCyE16voJ+w7vVNCdUhNZ9AmvOj\nsgrnO3XlDsgz20DymIAzXz6IzInpGr559SWaI2VgvoaDNcSJLJQ41090UiySVw/sx+lSzSs00vc+\nMnVS4lkztY6Rdc200Ntv+tFjtXryxfNNBnpHsJI7lcN4Jo8E7agPLC8tHedSFeEgQdBP+tCUr2Sc\n/UxPC8URzI/VVbVYA4lx7MiRZ82xjaWv14yshYW6CjwtAgdbuLHFd6yImeZzwOq3cxihnagCjwM8\ny8lJmgYSBP14d0xVlWQ/kXqyIVyoH6ubErVoQ7YFtyLAmokiskUja2GhrgJHyeN2LplMV7vMk8Jl\nzdb3j8uHbKsBl9ST5sdYiNFS6Z53SqUGPd/xjylrYD+JGk8w8Gmp9EypdL8STlfUpB82USHBmizH\nXSmZDhbqKqD9cLL1+plndJLLms3/vOc8E9lP5iLisybRQqnk9s8+MWw/7SaOqNldL6cratJPPuin\nXcVDJhMWFuoqIOuV5smbKRME8/NAC7trul6ZddVITfJaN5Bxrp9kXgyQ5Pys5nhV6XD7x+l6ohbq\nhzXbNchVIbIWFuoqIPtZronFSkb3D6qHeJA+rGDMN/v5KD5jSKTIh4EErp8R00+kqP208JRw3f3j\ndDmbY36oH9bcVhudZGSkGED2IwIL9MbWgN2V0zxfb2M0t9LEfck5RHPAaBmY8N5iSLzOh4EErp/U\nJTGENPNzWM1gvZyuqnGx3HqxZrxyHfVV1kKdn8P4gM49Gz7i4xXyv0x8DO/hr2qpT174gqZM55kD\nLaJYkw8DCVw/WNRrgmSns8bl59aTYdcr0BW19PLG2eWN04Vj5FkzM/GoRIqshbp/1iF39/Cx11Fy\npYaG28/e5xeoaEh7vhjl9k8ooQd2awayZ2HPl/y2DR+7HMWfzPDIBF/xetH+KvIyd/fP9+HQAbhb\nM5DtNI9XjcAtAxICl2wiGeYfA4N7nq1IIyAXFlo93HDD1ezKdj66AKiA/G80dHh5Q7m+NgVe+3JL\nd/4mYkuHq9mVff9J3U/M8nw+ho7thtk5NIDEluTARM837s2Z/wDFpogVXcYxwAAAAABJRU5ErkJg\ngg==\n", + "text/latex": [ + "$$(R \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} + \\dot{\\xi})\\mathbf{i}_{0} + R \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}\\mathbf{j}_{0}$$" + ], + "text/plain": [ + "⎛ d d ⎞\n", + "⎜R⋅cos(θ(t))⋅──(θ(t)) + ──(ξ(t))⎟\n", + "⎝ dt dt ⎠ 0_x + d \n", + "R⋅sin(θ(t))⋅──(θ(t))\n", + " dt 0_y" + ] + }, + "execution_count": 54, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Hallamos la velocidad de P en A1, expresada en A0\n", + "# ¡Con esta llamada ya estamos diciendo que C y P son fijos en A2!\n", + "P.v2pt_theory(C, A1, A2).express(A0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Misión cumplida :)**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "---" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "_Hemos hecho un repaso bastante profundo de las posibilidades del paquete `mechanics` de SymPy. Nos hemos dejado algunas cosas en el tintero pero no demasiadas: esta funcionalidad aún se está expandiendo y necesita pulir algunos detalles._\n", + "\n", + "**Referencias**\n", + "\n", + "* Capítulo de **aeromecánica** del libro de Cuerva y otros http://nbviewer.ipython.org/gist/Juanlu001/7711865\n", + "* Estabilidad longitudinal de un Boeing 747 http://nbviewer.ipython.org/github/AlexS12/Mecanica_Vuelo/blob/master/MVII_MatrizSistema.ipynb\n", + "\n", + "_¿Serás tú el siguiente que publique un notebook usando SymPy? ;)_" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Si te ha gustado esta clase:\n", + "\n", + "Tweet\n", + "\n", + "\n", + "---" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "####

¡Síguenos en Twitter!" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###### Follow @Pybonacci Follow @Alex__S12 Follow @newlawrence " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "##### \"Licencia
Curso AeroPython por Juan Luis Cano Rodriguez y Alejandro Sáez Mollejo se distribuye bajo una Licencia Creative Commons Atribución 4.0 Internacional." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "##### " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "---\n", + "_Las siguientes celdas contienen configuración del Notebook_\n", + "\n", + "_Para visualizar y utlizar los enlaces a Twitter el notebook debe ejecutarse como [seguro](http://ipython.org/ipython-doc/dev/notebook/security.html)_\n", + "\n", + " File > Trusted Notebook" + ] + }, + { + "cell_type": "code", + "execution_count": 55, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "Follow @Pybonacci\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "%%html\n", + "Follow @Pybonacci\n", + "" + ] + }, + { + "cell_type": "code", + "execution_count": 56, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "/* This template is inspired in the one used by Lorena Barba\n", + "in the numerical-mooc repository: https://github.com/numerical-mooc/numerical-mooc\n", + "We thank her work and hope you also enjoy the look of the notobooks with this style */\n", + "\n", + "\n", + "\n", + "El estilo se ha aplicado =)\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "execution_count": 56, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Esta celda da el estilo al notebook\n", + "from IPython.core.display import HTML\n", + "css_file = '../styles/aeropython.css'\n", + "HTML(open(css_file, \"r\").read())" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.5.4" + } + }, + "nbformat": 4, + "nbformat_minor": 1 +} diff --git a/notebooks_vacios/040-SymPy-Intro.ipynb b/notebooks_vacios/040-SymPy-Intro.ipynb index 20ca627..4bf15c6 100644 --- a/notebooks_vacios/040-SymPy-Intro.ipynb +++ b/notebooks_vacios/040-SymPy-Intro.ipynb @@ -11,7 +11,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# CLASE 5a: SymPy" + "# Introducción a SymPy" ] }, { @@ -20,7 +20,7 @@ "source": [ "![](http://sympy.org/static/images/logo.png)\n", "\n", - "_ __SymPy es una biblioteca de Python para matemática simbólica__. Apunta a convertirse en un sistema de algebra computacional (__CAS__) con todas sus prestaciones manteniendo el código tan simple como sea posible para manterlo comprensible y fácilmente extensible. SymPy está __escrito totalmente en Python y no requiere bibliotecas adicionales__. _Este proyecto comenzó en 2005, fue lanzado al público en 2007 y a él han contribuido durante estos años cientos de personas._\n", + "__SymPy es una biblioteca de Python para matemática simbólica__. Apunta a convertirse en un sistema de algebra computacional (__CAS__) con todas sus prestaciones manteniendo el código tan simple como sea posible para manterlo comprensible y fácilmente extensible. SymPy está __escrito totalmente en Python y no requiere bibliotecas adicionales__. _Este proyecto comenzó en 2005, fue lanzado al público en 2007 y a él han contribuido durante estos años cientos de personas._\n", "\n", "_ Otros CAS conocidos son Mathematica y Maple, sin embargo ambos son software privativo y de pago. [Aquí](https://github.com/sympy/sympy/wiki/SymPy-vs.-Maple) puedes encontrar una comparativa de SymPy con Maple. _\n", "\n", @@ -47,9 +47,7 @@ { "cell_type": "code", "execution_count": 1, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { @@ -57,7 +55,7 @@ "" ], "text/plain": [ - "" + "" ] }, "execution_count": 1, @@ -80,9 +78,7 @@ { "cell_type": "code", "execution_count": 2, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [ { "data": { @@ -90,7 +86,7 @@ "" ], "text/plain": [ - "" + "" ] }, "execution_count": 2, @@ -119,9 +115,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Importación" @@ -130,9 +124,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -158,9 +150,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Intentamos usar un símbolo que no hemos creado" @@ -178,9 +168,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos el símbolo a" @@ -189,9 +177,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Número pi" @@ -200,9 +186,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Unidad imaginaria" @@ -211,9 +195,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Número e" @@ -222,9 +204,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Vemos qué tipo de variable es a" @@ -240,18 +220,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -267,18 +243,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -306,9 +278,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#creación de símbolos" @@ -324,9 +294,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Diferencia entre variable y símbolo" @@ -342,9 +310,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -358,9 +324,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -382,9 +346,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos símbolos reales" @@ -393,9 +355,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Podemos ver las asunciones de un símbolo" @@ -418,9 +378,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -441,9 +399,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -471,18 +427,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Sustituimos x por y ** 2" @@ -491,9 +443,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# ¡Pero la expresión no cambia!" @@ -502,9 +452,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Para que cambie" @@ -520,9 +468,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -536,9 +482,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -552,18 +496,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#ver pi con 25 decimales" @@ -572,9 +512,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#el mismo resultado se obtiene ocn la función N()" @@ -633,9 +571,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#1" @@ -644,18 +580,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#2" @@ -664,18 +596,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#3" @@ -684,18 +612,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#4" @@ -704,27 +628,21 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -745,9 +663,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#creamos una expresión\n", @@ -757,9 +673,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#utilizando método" @@ -775,18 +689,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -800,18 +710,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -825,9 +731,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -841,9 +745,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos una función F" @@ -852,9 +754,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos una función G" @@ -870,9 +770,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Derivamos la función compuesta F(G(x))" @@ -888,9 +786,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# definimos una f" @@ -899,9 +795,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# definimos una g(f)" @@ -910,9 +804,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#la derivamos" @@ -940,27 +832,21 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -985,9 +871,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -1001,9 +885,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -1024,9 +906,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#creamos la expresión" @@ -1035,9 +915,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#la desarrollamos en serie" @@ -1053,9 +931,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Indicando el número de términos" @@ -1071,18 +947,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -1110,9 +982,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#creamos la ecuación" @@ -1121,9 +991,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# También la podemos crear como" @@ -1132,9 +1000,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#la resolvemos" @@ -1151,9 +1017,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos los símbolos y la ecuación" @@ -1162,9 +1026,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# La resolvemos" @@ -1196,18 +1058,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#resolvemos" @@ -1223,9 +1081,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#creamos una matriz llena de símbolos" @@ -1234,9 +1090,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#sacamos autovalores" @@ -1245,9 +1099,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#inversa" @@ -1256,9 +1108,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#elevamos al cuadrado la matriz" @@ -1328,22 +1178,22 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false - }, + "execution_count": 77, + "metadata": {}, "outputs": [ { "data": { "text/html": [ + "/* This template is inspired in the one used by Lorena Barba\n", + "in the numerical-mooc repository: https://github.com/numerical-mooc/numerical-mooc\n", + "We thank her work and hope you also enjoy the look of the notobooks with this style */\n", + "\n", "\n", "\n", "El estilo se ha aplicado =)\n", @@ -1421,7 +1271,7 @@ " font-weight: 600;\n", " font-size: 35pt;\n", " line-height: 100%;\n", - " color: #234765;\n", + " color: #000000;\n", " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " display: block;\n", @@ -1434,7 +1284,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #234765;\n", + " color: #2c6391;\n", "}\t\n", "\n", ".text_cell_render h3 {\n", @@ -1444,7 +1294,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #376f9e;\n", + " color: #387eb8;\n", "}\n", "\n", ".text_cell_render h4 { /*Use this for captions*/\n", @@ -1454,7 +1304,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #376f9e;\n", + " color: #5797cc;\n", "}\n", "\n", ".text_cell_render h5 { /*Use this for small titles*/\n", @@ -1503,10 +1353,10 @@ "\n" ], "text/plain": [ - "" + "" ] }, - "execution_count": 1, + "execution_count": 77, "metadata": {}, "output_type": "execute_result" } @@ -1535,9 +1385,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.3" + "version": "3.5.4" } }, "nbformat": 4, - "nbformat_minor": 0 + "nbformat_minor": 1 } diff --git a/notebooks_vacios/040-SymPy-Mecanica.ipynb b/notebooks_vacios/041-SymPy-Mecanica.ipynb similarity index 89% rename from notebooks_vacios/040-SymPy-Mecanica.ipynb rename to notebooks_vacios/041-SymPy-Mecanica.ipynb index e67009c..efb1252 100644 --- a/notebooks_vacios/040-SymPy-Mecanica.ipynb +++ b/notebooks_vacios/041-SymPy-Mecanica.ipynb @@ -11,7 +11,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# Clase 5b: Mecánica con SymPy" + "# Mecánica con SymPy" ] }, { @@ -40,9 +40,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -58,18 +56,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -83,9 +77,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -102,9 +94,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -117,21 +107,24 @@ }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, + "execution_count": 6, + "metadata": {}, "outputs": [], "source": [ - "# Definimos nuestra propia clase para que los versores sean IJK" + "# Definimos nuestra propia clase para que los versores sean IJK\n", + "# aeropython: preserve\n", + "class IJKReferenceFrame(ReferenceFrame):\n", + " def __init__(self, name):\n", + " super().__init__(name, latexs=['\\mathbf{%s}_{%s}' % (idx, name) for idx in (\"i\", \"j\", \"k\")])\n", + " self.i = self.x\n", + " self.j = self.y\n", + " self.k = self.z" ] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -152,54 +145,42 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -213,18 +194,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -244,45 +221,35 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -311,18 +278,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -356,18 +319,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -383,18 +342,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -408,9 +363,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -430,63 +383,49 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -502,27 +441,21 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -538,18 +471,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "#v1.diff(dynamicsymbols._t, A2)" @@ -558,18 +487,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -590,18 +515,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -615,9 +536,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -631,9 +550,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -647,9 +564,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -685,9 +600,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, @@ -707,9 +620,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos nuestros sistemas de referencia" @@ -718,9 +629,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos los símbolos dinámicos necesarios" @@ -729,9 +638,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Orientamos los sistemas de referencia" @@ -740,18 +647,14 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Creamos el punto C, centro del disco, y especificamos su velocidad\n", @@ -761,9 +664,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Localizamos el punto P, punto fijo del disco, respecto a C, en\n", @@ -773,9 +674,7 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [ "# Hallamos la velocidad de P en A1, expresada en A0\n", @@ -865,22 +764,22 @@ { "cell_type": "code", "execution_count": null, - "metadata": { - "collapsed": false - }, + "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false - }, + "execution_count": 56, + "metadata": {}, "outputs": [ { "data": { "text/html": [ + "/* This template is inspired in the one used by Lorena Barba\n", + "in the numerical-mooc repository: https://github.com/numerical-mooc/numerical-mooc\n", + "We thank her work and hope you also enjoy the look of the notobooks with this style */\n", + "\n", "\n", "\n", "El estilo se ha aplicado =)\n", @@ -958,7 +857,7 @@ " font-weight: 600;\n", " font-size: 35pt;\n", " line-height: 100%;\n", - " color: #234765;\n", + " color: #000000;\n", " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " display: block;\n", @@ -971,7 +870,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #234765;\n", + " color: #2c6391;\n", "}\t\n", "\n", ".text_cell_render h3 {\n", @@ -981,7 +880,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #376f9e;\n", + " color: #387eb8;\n", "}\n", "\n", ".text_cell_render h4 { /*Use this for captions*/\n", @@ -991,7 +890,7 @@ " margin-bottom: 0.1em;\n", " margin-top: 0.3em;\n", " font-style: regular;\n", - " color: #376f9e;\n", + " color: #5797cc;\n", "}\n", "\n", ".text_cell_render h5 { /*Use this for small titles*/\n", @@ -1040,10 +939,10 @@ "\n" ], "text/plain": [ - "" + "" ] }, - "execution_count": 1, + "execution_count": 56, "metadata": {}, "output_type": "execute_result" } @@ -1072,9 +971,9 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.4.3" + "version": "3.5.4" } }, "nbformat": 4, - "nbformat_minor": 0 + "nbformat_minor": 1 } From a5cd3cc2df8ad763f6795da00825a8a9ab74a376 Mon Sep 17 00:00:00 2001 From: AunSiro Date: Thu, 22 Mar 2018 12:32:37 +0100 Subject: [PATCH 2/2] Error de init_session en notebook rodeado MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Se importan las funciones y variables de manera diferente para que funcione mientras no esté disponible init_session --- notebooks_completos/040-SymPy-Intro.ipynb | 488 +++++++++---------- notebooks_completos/041-SymPy-Mecanica.ipynb | 431 ++++++++-------- notebooks_vacios/040-SymPy-Intro.ipynb | 41 +- notebooks_vacios/041-SymPy-Mecanica.ipynb | 46 +- 4 files changed, 534 insertions(+), 472 deletions(-) diff --git a/notebooks_completos/040-SymPy-Intro.ipynb b/notebooks_completos/040-SymPy-Intro.ipynb index 2a20d8b..771a421 100644 --- a/notebooks_completos/040-SymPy-Intro.ipynb +++ b/notebooks_completos/040-SymPy-Intro.ipynb @@ -109,7 +109,24 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Lo primero, como siempre, es importar aquello que vayamos a necesitar:" + "Lo primero, como siempre, es importar aquello que vayamos a necesitar. La manera usual de hacerlo con SymPy es importar la función `init_session`:\n", + "```\n", + "from sympy import init_session\n", + "init_session(use_latex=True)```\n", + "\n", + " Esta función ya se encarga de importar todas las funciones básicas y preparar las salidas gráficas. Sin embargo, en este momento, esta función se encuentra en mantenimiento para su uso dentro de los notebooks por lo que activaremos la salida gráfica e importaremos las funciones de la manera usual. Puedes consultar el estado de la corrección en: https://github.com/sympy/sympy/pull/13300 y https://github.com/sympy/sympy/issues/13319 ." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "El comando `init_session` llevaría a cabo algunas acciones por nostros:\n", + "\n", + "* Gracias a `use_latex=True` obtenemos la salida en $\\LaTeX$.\n", + "* __Crea una serie de variables__ para que podamos ponernos a trabajar en el momento.\n", + "\n", + "Estas capacidades volverán a estar disponibles cuando el problema se corrija." ] }, { @@ -118,45 +135,28 @@ "metadata": {}, "outputs": [], "source": [ - "# Importación\n", - "from sympy import init_session" + "from sympy import init_printing" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "IPython console for SymPy 1.0 (Python 3.5.4-64-bit) (ground types: python)\n", - "\n", - "These commands were executed:\n", - ">>> from __future__ import division\n", - ">>> from sympy import *\n", - ">>> x, y, z, t = symbols('x y z t')\n", - ">>> k, m, n = symbols('k m n', integer=True)\n", - ">>> f, g, h = symbols('f g h', cls=Function)\n", - ">>> init_printing()\n", - "\n", - "Documentation can be found at http://docs.sympy.org/1.0/\n" - ] - } - ], + "outputs": [], "source": [ - "init_session(use_latex=True)" + "init_printing() " ] }, { - "cell_type": "markdown", + "cell_type": "code", + "execution_count": 5, "metadata": {}, + "outputs": [], "source": [ - "Lo primero que vemos es que el comando `init_session` ha llevado a cabo algunas acciones por nostros:\n", - "\n", - "* Gracias a `use_latex=True` obtenemos la salida en $\\LaTeX$.\n", - "* __Ha creado una serie de variables__ para que podamos ponernos a trabajar en el momento." + "# aeropython: preserve\n", + "from sympy import (symbols, pi, I, E, cos, sin, exp, tan, simplify, expand, factor, collect,\n", + " apart, cancel, expand_trig, diff, Derivative, Function, integrate, limit,\n", + " series, Eq, solve, dsolve, Matrix, N)" ] }, { @@ -170,7 +170,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 6, "metadata": {}, "outputs": [ { @@ -180,7 +180,7 @@ "traceback": [ "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[1;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 1\u001b[0m \u001b[1;31m# Intentamos usar un símbolo que no hemos creado\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0ma\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m2\u001b[0m \u001b[1;33m*\u001b[0m \u001b[0mb\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m", + "\u001b[1;32m\u001b[0m in \u001b[0;36m\u001b[1;34m()\u001b[0m\n\u001b[0;32m 1\u001b[0m \u001b[1;31m# Intentamos usar un símbolo que no hemos creado\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0ma\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m2\u001b[0m \u001b[1;33m*\u001b[0m \u001b[0mb\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[1;31mNameError\u001b[0m: name 'b' is not defined" ] } @@ -201,12 +201,12 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 7, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAARklEQVQIHWNgYFRWCWNgYCgV4PjAwMAWycC0\ngIGB6RsDtwMDA+sChvwDDAz5Bgz3GYDUBYYVvAIMHA6c+5iBKp/YyLkzAABmaAxT1tVPwQAAAABJ\nRU5ErkJggg==\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAABHNCSVQICAgIfAhkiAAAANhJREFUKJHN0TFKQ0EYBOAPopBCQQkSLVJ4AcHulV4jFoJ1zpDG0jvYKAHBC6ggVkpIiJBOU5ouEVNYKISgxdvg8rJNwMIftpnZ+Wdmlz+aQ7QwwAcm6OJ4fqFUEKyjjU/0cI9XZKjjCw9FlzVsJ9x3guvzsrFfMEoRm2iGiBPM8B2dJ1iJBHu4RRUdXOIdU+ziCP2iS19e9CCR4CQ4NWKwFsDrhGADw8BnMbEVwAFWI7yCm8BNUS5uvPNb9hQXeMOV/EEW+szdzjGW/8mjvPx+WHaWEv3D+QGVVDVPL1hNBgAAAABJRU5ErkJggg==\n", "text/latex": [ "$$a$$" ], @@ -214,7 +214,7 @@ "a" ] }, - "execution_count": 6, + "execution_count": 7, "metadata": {}, "output_type": "execute_result" } @@ -227,12 +227,12 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 8, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAbBAMAAADc0o+gAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABSkAAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$\\left(a + \\pi\\right)^{2}$$" ], @@ -241,7 +241,7 @@ "(a + π) " ] }, - "execution_count": 7, + "execution_count": 8, "metadata": {}, "output_type": "execute_result" } @@ -253,12 +253,12 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 9, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAQBAMAAABEqSrGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4UlEQVQYGWNgwARCWmuAgq8wJRgY2B4w2Bcw\nMLxEkbsC4fElMLAGoEgAOVA51g0MfH9wyHF8g8sxKquEQVRB9QE53N/YnByAdKkAxwcgBQQIuXoD\nZnaglWyRDEwLwFJIcvsYZJkaGBiYvjFwOwDlLDo6tnZ09IFVMSkwJPADBVkXMOQfYAADuJlOQO79\nBAaGfAOG+xApuJm8CgyCDH1sQLkLDCt4BVD0HWJgcGT7y8zAwOHAuQ9IgQDUTMbNxnob2D9IAt35\nxEbOHUWO4////xsYXk6ACEJJuFtQRCGcRDQxAA11NDjP+3KqAAAAAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$a + 2 i$$" ], @@ -266,7 +266,7 @@ "a + 2⋅ⅈ" ] }, - "execution_count": 8, + "execution_count": 9, "metadata": {}, "output_type": "execute_result" } @@ -278,12 +278,12 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 10, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAJBAMAAAASvxsjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\nkgMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAA4SURBVAgdY2BgVDYJYGBID2BVYGDtZBBZwMDdY3yQ\ngYFvAwMQ8CmASA4gacDAfIFBBKjSxOYAAwDWpwf9jP6jxwAAAABJRU5ErkJggg==\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAAPCAYAAADUFP50AAAABHNCSVQICAgIfAhkiAAAAMtJREFUKJHN0r9KgmEUx/GPWdAFtAlegLRFtLgZXkBegYubN9Dk1toQRKOTi4NLtLg4eAHmJN6ASLNDQTm8Z3h466GIhn7w8MA553v+POfhj9XBE17wijWuUc0BVYzwEcH3uMUqbMMceBcBNzhM7EeYh69Rhi7wjkkmaS/ArlLWPirYYfAFeBp3pezYRsbvzmUKHYdxlmnzkw5K5U9+CqZaRNWrjL8p2WM6aBuPigeb4jk6quFMsZJ6ruo5xtjgTfFzlnhA6zdj/APtAaO+KwMQliPnAAAAAElFTkSuQmCC\n", "text/latex": [ "$$e$$" ], @@ -291,7 +291,7 @@ "ℯ" ] }, - "execution_count": 9, + "execution_count": 10, "metadata": {}, "output_type": "execute_result" } @@ -303,7 +303,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 11, "metadata": {}, "outputs": [ { @@ -312,7 +312,7 @@ "sympy.core.symbol.Symbol" ] }, - "execution_count": 10, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } @@ -331,12 +331,12 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 12, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAgUlEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA9gECo6GQBkcXxn4JjIsB+kiCuBIfIBgzKIqcTAoB/AsAbI4lFgYJjMwPoF\nyLzEwCj6nYH3AwMDiwIDj+hfBuYNFxmkyouyGPYwvEooYNj///8vhrDjjjkNAMjNHzXk+S3KAAAA\nAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$2 a$$" ], @@ -344,7 +344,7 @@ "2⋅a" ] }, - "execution_count": 11, + "execution_count": 12, "metadata": {}, "output_type": "execute_result" } @@ -356,7 +356,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 13, "metadata": {}, "outputs": [ { @@ -365,7 +365,7 @@ "sympy.core.mul.Mul" ] }, - "execution_count": 12, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } @@ -385,12 +385,12 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 14, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAPBAMAAABJi79qAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABPklEQVQYGYWQvUoDQRSFv8WMG7MRU1uFFP5g\nM5DKyu0s3SogFgbBgDYJQbQRNi8gYiOo5AkstBK7BLERC0V8gLWwEIUggpGkWO/MxNop9hzO+WaZ\ne/FK5RA5Tv39HfGZImp2tzDKKvg/hrCqrlkQn9OsJt6Ry9iEE0NYHStSFn+n2YKNUXcK9VBSq4cF\ncXjvmiHELqMTOcLqogHwA62+4CH867iIbCE6XL5pwUugOZN/iHVdtu+AbF8NE9qorhAxdBJwXa7q\niFxVpRHzkY8QQXcyfpSpbFdyAKID6LWeDcHHbRxiMsgXzdfpuRCvDUvIiNGoe8KzMxqVLfQOarX6\ncVeuXIHtZMN5Q1hdkXeIn9C8FTKfLmN6r7nN+qXT8YZqCzGluY/WEpfRSdOBebJVmjOhbGzpO/FL\nczKv6f4/v2PcYrROz5eyAAAAAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$2.26492$$" ], @@ -398,7 +398,7 @@ "2.26492" ] }, - "execution_count": 13, + "execution_count": 14, "metadata": {}, "output_type": "execute_result" } @@ -410,7 +410,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 15, "metadata": {}, "outputs": [ { @@ -419,7 +419,7 @@ "float" ] }, - "execution_count": 14, + "execution_count": 15, "metadata": {}, "output_type": "execute_result" } @@ -451,12 +451,12 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 16, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAMBAMAAAB/4Ov2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uze+rmRC7\nZkTTotXTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNgYBBSMmCAAtcE9oIbs87Pmn+B\ngaGRgTNhNsN9Bt4HDFy/GRi4FRjWMLA5MHAcYGBg3sAQyMDHwLBfAKSP+x+I5AdqSWDgA6pkYOCa\nwLDLgIExAMRmELyrwMDAMwHMBhMglTBgbwBjMbyMnwhkAwDTtxfeCPtWNAAAAABJRU5ErkJggg==\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAATCAYAAAB7u5a2AAAABHNCSVQICAgIfAhkiAAAAQpJREFUOI3t080rRUEcxvHPJSKKUt5KFsqaa32L1N3YsKDk/2FjYecPsCIrJeFPkGzYyUKk5CUrSV4WZ9TtOGe4p242nppmzm+e53um5jf8gSaxiSu84Ab7mK0H0pxRW8M6+gLwEI+o4C58F9IyPrCNjtReJ4aLgsfxhiO0FoXkaUNy6umIpxo8sbGYFbzFA5oi8Hb014xrrKZqLelQW/jrSQScVk/IzOUZvk5ZCnNvHfBymI9/gj/jFINYyPCN+t62E7jHZR68VLOewU6o7eEM3RjDEAZS2a2wX82Dp1XBruRiXyUv8wBLGd5zrPwWXI+68I75RsCnJJ0yEjPFejqmMp5wUTD/rwbpE9FtNMbHngAmAAAAAElFTkSuQmCC\n", "text/latex": [ "$$c_{T}$$" ], @@ -464,7 +464,7 @@ "c_T" ] }, - "execution_count": 15, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } @@ -484,12 +484,12 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 17, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAELvv3c2rVESJdpkiZvtO\n0HUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABSSURBVAgdY2BUdmAAAtYEEMlXACL7BEDkFBDBELZq\nOwMD41MBuQMM7A8Z+CYwcAcwyE1gYN3A0FfAwHeBYZ8AA18Doy0DA9sFpg1Albv2MjAAAB2yD7/K\nbXugAAAAAElFTkSuQmCC\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAATCAYAAACgADyUAAAABHNCSVQICAgIfAhkiAAAAPpJREFUOI3V0rtKQ0EQxvGfmiiohdipRV7A4K2wtRHS6iPYCFbWPoZlwMbGygtYGcHOQvFSphAk9rFRRBRFi7MH4rInErBxYBmYmT/zzczyh7aBL6x1K+pPxOaDv+614y1eUeoFGsI7Ln4rjKVWUZbJnMU+HvGCBqaLwIXgKziXLWkHN1jGGcZSCuqhuI25KLcbclsp8ErxKRZDbi9OlPGGlvRGKwE85OeMVQziGB8FIDzEYH74VgKCleAbMZhvdDwBTWAddznYaZeyGe4x0hEfxSk+UcuDfcGX8IwmhjGAI9lPWsUkNrEdd5sJ3eqYwgGewjvBUkL+f7FvAyYxszVLv64AAAAASUVORK5CYII=\n", "text/latex": [ "$$b$$" ], @@ -497,7 +497,7 @@ "b" ] }, - "execution_count": 16, + "execution_count": 17, "metadata": {}, "output_type": "execute_result" } @@ -517,7 +517,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 18, "metadata": {}, "outputs": [], "source": [ @@ -533,12 +533,12 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 19, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAUBAMAAAAthD11AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQ\nMqvFy5UvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABSElEQVQoFX2SPUsDQRCGn7tcPsglISAKYnN2\ndlF74SwshfsJsRHLNBZWnpWtlYpVKgsrQYLGSrAQq+Qf5A9YaIwoCInvRkgWL+fADe/sPLe7M7Pg\nBKSYc3ykzLLJzqUgFLeq89tQqQrYTINqARzGlEJwoxSoPFDCe4NbyJndZln2W6v+JzzDwizArHlf\ncu67RJN9yVV9i0052zJ9RfkhZCPWhTcVdho2YfIfchWRbsieUEX4sfGWlU4V5E6gUOcCriDETRRQ\n00ovhvKrgRqUI3xrk1/p6QI3koJ0XEQ+Zukv5Dgh7DhQrJuLb9CLko33ecGJdIAb0oJ263zYDiiZ\nFk+tsnJP4U4dyl5zCY+D7sGZsrsPU0Tljvq4I3Xc69pjKSQKHP+kmVgDThY4hjRg1iZnPE2ULcxT\nsR5dYOcmOmMu4QRy/5ie7w8XTDqYWIHfzwAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$\\left ( \\omega, \\quad \\Omega\\right )$$" ], @@ -546,7 +546,7 @@ "(ω, Ω)" ] }, - "execution_count": 18, + "execution_count": 19, "metadata": {}, "output_type": "execute_result" } @@ -574,7 +574,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 20, "metadata": {}, "outputs": [], "source": [ @@ -584,7 +584,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 21, "metadata": {}, "outputs": [ { @@ -597,7 +597,7 @@ " 'real': True}" ] }, - "execution_count": 20, + "execution_count": 21, "metadata": {}, "output_type": "execute_result" } @@ -623,12 +623,12 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 22, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAZBAMAAAAyHoooAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$\\sin^{2}{\\left (x \\right )} + \\cos^{2}{\\left (x \\right )}$$" ], @@ -637,7 +637,7 @@ "sin (x) + cos (x)" ] }, - "execution_count": 21, + "execution_count": 22, "metadata": {}, "output_type": "execute_result" } @@ -663,12 +663,12 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 23, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFUKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSxoPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA6XoX52TB9a4AAAAASUVORK5CYII=\n", "text/latex": [ "$$1$$" ], @@ -676,7 +676,7 @@ "1" ] }, - "execution_count": 22, + "execution_count": 23, "metadata": {}, "output_type": "execute_result" } @@ -708,12 +708,12 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 24, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\sin^{2}{\\left (x \\right )} + \\cos^{2}{\\left (x \\right )}$$" ], @@ -722,7 +722,7 @@ "sin (x) + cos (x)" ] }, - "execution_count": 23, + "execution_count": 24, "metadata": {}, "output_type": "execute_result" } @@ -733,12 +733,12 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 25, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\sin^{2}{\\left (y^{2} \\right )} + \\cos^{2}{\\left (y^{2} \\right )}$$" ], @@ -747,7 +747,7 @@ "sin ⎝y ⎠ + cos ⎝y ⎠" ] }, - "execution_count": 24, + "execution_count": 25, "metadata": {}, "output_type": "execute_result" } @@ -759,12 +759,12 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 26, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\sin^{2}{\\left (x \\right )} + \\cos^{2}{\\left (x \\right )}$$" ], @@ -773,7 +773,7 @@ "sin (x) + cos (x)" ] }, - "execution_count": 25, + "execution_count": 26, "metadata": {}, "output_type": "execute_result" } @@ -785,12 +785,12 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 27, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\sin^{2}{\\left (y^{2} \\right )} + \\cos^{2}{\\left (y^{2} \\right )}$$" ], @@ -799,7 +799,7 @@ "sin ⎝y ⎠ + cos ⎝y ⎠" ] }, - "execution_count": 26, + "execution_count": 27, "metadata": {}, "output_type": "execute_result" } @@ -819,12 +819,12 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 28, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\sin^{2}{\\left (y^{2} \\right )} + \\cos^{2}{\\left (y^{2} \\right )}$$" ], @@ -833,7 +833,7 @@ "sin ⎝y ⎠ + cos ⎝y ⎠" ] }, - "execution_count": 27, + "execution_count": 28, "metadata": {}, "output_type": "execute_result" } @@ -851,12 +851,12 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 29, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAOBAMAAADZH/H+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiUlEQVQIHWNgEFIyYYACxgAG/wQom/0LA38D\nlM25kmH/ASgbSMHVMDBwNzEw1P////8rAwOjRjQDk667WKEASKXWhCsMiuwXwJq4VjMwGPBtACoR\nYGD+wsD7gAMozP8bxGZN4AeKcxkwcHxk2L+BfwIDA1sCg38BQwUDlwJQ0dRQSwYGMwY+BQYAQRkc\nPIr4GKkAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$3 \\pi$$" ], @@ -864,7 +864,7 @@ "3⋅π" ] }, - "execution_count": 28, + "execution_count": 29, "metadata": {}, "output_type": "execute_result" } @@ -882,12 +882,12 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 30, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$9.42477796076937971538793$$" ], @@ -895,7 +895,7 @@ "9.424777960769379715387930" ] }, - "execution_count": 29, + "execution_count": 30, "metadata": {}, "output_type": "execute_result" } @@ -906,12 +906,12 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 31, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$3.141592653589793238462643$$" ], @@ -919,7 +919,7 @@ "3.141592653589793238462643" ] }, - "execution_count": 30, + "execution_count": 31, "metadata": {}, "output_type": "execute_result" } @@ -931,12 +931,12 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 32, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$3.141592653589793238462643$$" ], @@ -944,7 +944,7 @@ "3.141592653589793238462643" ] }, - "execution_count": 31, + "execution_count": 32, "metadata": {}, "output_type": "execute_result" } @@ -1006,12 +1006,12 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 33, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\left(x^{3} + 3 y + 2\\right)^{2}$$" ], @@ -1021,7 +1021,7 @@ "⎝x + 3⋅y + 2⎠ " ] }, - "execution_count": 32, + "execution_count": 33, "metadata": {}, "output_type": "execute_result" } @@ -1034,12 +1034,12 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 34, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$x^{6} + 6 x^{3} y + 4 x^{3} + 9 y^{2} + 12 y + 4$$" ], @@ -1048,7 +1048,7 @@ "x + 6⋅x ⋅y + 4⋅x + 9⋅y + 12⋅y + 4" ] }, - "execution_count": 33, + "execution_count": 34, "metadata": {}, "output_type": "execute_result" } @@ -1060,12 +1060,12 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 35, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\frac{1}{\\left(x - 1\\right)^{2}} \\left(3 x^{2} - 2 x + 1\\right)$$" ], @@ -1077,7 +1077,7 @@ " (x - 1) " ] }, - "execution_count": 34, + "execution_count": 35, "metadata": {}, "output_type": "execute_result" } @@ -1090,12 +1090,12 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 36, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": "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\n", "text/latex": [ "$$3 + \\frac{4}{x - 1} + \\frac{2}{\\left(x - 1\\right)^{2}}$$" ], @@ -1106,7 +1106,7 @@ " (x - 1) " ] }, - "execution_count": 35, + "execution_count": 36, "metadata": {}, "output_type": "execute_result" } @@ -1117,12 +1117,12 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 37, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAAWBAMAAACfwnvZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8B/JE6faQ30O2wAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$x^{3} + 9 x^{2} + 27 x + 27$$" ], @@ -1131,7 +1131,7 @@ "x + 9⋅x + 27⋅x + 27" ] }, - "execution_count": 36, + "execution_count": 37, "metadata": {}, "output_type": "execute_result" } @@ -1144,12 +1144,12 @@ }, { "cell_type": "code", - "execution_count": 37, + "execution_count": 38, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAbBAMAAAAzEOSeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$\\left(x + 3\\right)^{3}$$" ], @@ -1158,7 +1158,7 @@ "(x + 3) " ] }, - "execution_count": 37, + "execution_count": 38, "metadata": {}, "output_type": "execute_result" } @@ -1169,12 +1169,12 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 39, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAUBAMAAABmLO6YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWQFPcVQAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (x + 2 y \\right )}$$" ], @@ -1182,7 +1182,7 @@ "sin(x + 2⋅y)" ] }, - "execution_count": 38, + "execution_count": 39, "metadata": {}, "output_type": "execute_result" } @@ -1195,12 +1195,12 @@ }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 40, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAUBAMAAABmLO6YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWQFPcVQAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (x + 2 y \\right )}$$" ], @@ -1208,7 +1208,7 @@ "sin(x + 2⋅y)" ] }, - "execution_count": 39, + "execution_count": 40, "metadata": {}, "output_type": "execute_result" } @@ -1219,12 +1219,12 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 41, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\left(2 \\cos^{2}{\\left (y \\right )} - 1\\right) \\sin{\\left (x \\right )} + 2 \\sin{\\left (y \\right )} \\cos{\\left (x \\right )} \\cos{\\left (y \\right )}$$" ], @@ -1233,7 +1233,7 @@ "⎝2⋅cos (y) - 1⎠⋅sin(x) + 2⋅sin(y)⋅cos(x)⋅cos(y)" ] }, - "execution_count": 40, + "execution_count": 41, "metadata": {}, "output_type": "execute_result" } @@ -1244,12 +1244,12 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 42, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$2 \\sin{\\left (x \\right )} \\cos^{2}{\\left (y \\right )} - \\sin{\\left (x \\right )} + 2 \\sin{\\left (y \\right )} \\cos{\\left (x \\right )} \\cos{\\left (y \\right )}$$" ], @@ -1258,7 +1258,7 @@ "2⋅sin(x)⋅cos (y) - sin(x) + 2⋅sin(y)⋅cos(x)⋅cos(y)" ] }, - "execution_count": 41, + "execution_count": 42, "metadata": {}, "output_type": "execute_result" } @@ -1283,12 +1283,12 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 43, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnZUZrtE7yKJ\nmatywa77AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQklEQVQoFWNgwAm4CyBS7BvQlXgFIER2wJiv\nYQwY7X8AxmJg0IYxhWAMLDT7BZggD5IFMDEYzQx3DuMCmBiU3uIAF9gMZzFoIpgMrulB3P0PWLSu\nhD4FicYCsWjczhigdXcRqhgbGJgZmB4wMLcy7AdZ1s7AwHiA3ZRZgYGhCKGKcyEDKwPHAwaOBQz8\nBUBhMwYGHgGej2xAHckMQsYgoAIUtm/dAFalwMAPtIVhJdAsBqYGIIvBH0RAgYz+X7CqBrgqBggL\nWRW3AEP9BJCNUFVAG4GmOIDIySACAtgCGPgKkFQBXc8mEM/AKoDieraFDBwbgAbBzDrLwDA/oZ9h\nO9CMOzCTGBg4j4efZHn/T+T9P7f+dUBhoDVyoaLXQd4F+Q0HQMQQtwEOJUBh4mKboRtmAr6UwwBP\nhXDlMG3INFKKBgDEpkPH0/6h/gAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$- \\sin{\\left (x \\right )}$$" ], @@ -1296,7 +1296,7 @@ "-sin(x)" ] }, - "execution_count": 42, + "execution_count": 43, "metadata": {}, "output_type": "execute_result" } @@ -1311,12 +1311,12 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 44, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnZUZrtE7yKJ\nmatywa77AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQklEQVQoFWNgwAm4CyBS7BvQlXgFIER2wJiv\nYQwY7X8AxmJg0IYxhWAMLDT7BZggD5IFMDEYzQx3DuMCmBiU3uIAF9gMZzFoIpgMrulB3P0PWLSu\nhD4FicYCsWjczhigdXcRqhgbGJgZmB4wMLcy7AdZ1s7AwHiA3ZRZgYGhCKGKcyEDKwPHAwaOBQz8\nBUBhMwYGHgGej2xAHckMQsYgoAIUtm/dAFalwMAPtIVhJdAsBqYGIIvBH0RAgYz+X7CqBrgqBggL\nWRW3AEP9BJCNUFVAG4GmOIDIySACAtgCGPgKkFQBXc8mEM/AKoDieraFDBwbgAbBzDrLwDA/oZ9h\nO9CMOzCTGBg4j4efZHn/T+T9P7f+dUBhoDVyoaLXQd4F+Q0HQMQQtwEOJUBh4mKboRtmAr6UwwBP\nhXDlMG3INFKKBgDEpkPH0/6h/gAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$- \\sin{\\left (x \\right )}$$" ], @@ -1324,7 +1324,7 @@ "-sin(x)" ] }, - "execution_count": 43, + "execution_count": 44, "metadata": {}, "output_type": "execute_result" } @@ -1343,12 +1343,12 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 45, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAUBAMAAADbzbjtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZUZs0Qu91E7yKJ\nmaurDqYVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKUlEQVQoFWNgQAdboALeUJpxEUIFdwGUzbcB\nwmD8g5BkhYoxsAUgBGEsXxiD4RCcBWd0w1lZEBa3IFyEuwHOZJ3AwKjsFLD9H4PoWiE9BaA40wUG\nBjal15OAZvJeYEhnYGhgiGNgaFfg/g2UZHdgYMhh8FBoZWBgCWCYlcCgwFDHwFDFwPAFKMlhwMBw\njOF8gjUDA3MDA8d3bQaQZCkDwyegJNcEBoYEhnAgi4HxAwPb0f8XQJLlCEkICyS5jYHlJ5IkyFgG\nxo8gnZwNDEYMDE+QJEEOMmP6wKAGdpBVAkM1yEyosTwXGDg/s3/gBqrhPcAwx8lngtz/Z3L/H5r/\negA2i+2Rmq4L0FhWASCBChDBtwNVAsTDG/DsG6AasEUZPLLhqpBNR0omACUOR3UP731RAAAAAElF\nTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (x \\right )}$$" ], @@ -1356,7 +1356,7 @@ "sin(x)" ] }, - "execution_count": 44, + "execution_count": 45, "metadata": {}, "output_type": "execute_result" } @@ -1367,12 +1367,12 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 46, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAUBAMAAADbzbjtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZUZs0Qu91E7yKJ\nmaurDqYVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKUlEQVQoFWNgQAdboALeUJpxEUIFdwGUzbcB\nwmD8g5BkhYoxsAUgBGEsXxiD4RCcBWd0w1lZEBa3IFyEuwHOZJ3AwKjsFLD9H4PoWiE9BaA40wUG\nBjal15OAZvJeYEhnYGhgiGNgaFfg/g2UZHdgYMhh8FBoZWBgCWCYlcCgwFDHwFDFwPAFKMlhwMBw\njOF8gjUDA3MDA8d3bQaQZCkDwyegJNcEBoYEhnAgi4HxAwPb0f8XQJLlCEkICyS5jYHlJ5IkyFgG\nxo8gnZwNDEYMDE+QJEEOMmP6wKAGdpBVAkM1yEyosTwXGDg/s3/gBqrhPcAwx8lngtz/Z3L/H5r/\negA2i+2Rmq4L0FhWASCBChDBtwNVAsTDG/DsG6AasEUZPLLhqpBNR0omACUOR3UP731RAAAAAElF\nTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (x \\right )}$$" ], @@ -1380,7 +1380,7 @@ "sin(x)" ] }, - "execution_count": 45, + "execution_count": 46, "metadata": {}, "output_type": "execute_result" } @@ -1398,12 +1398,12 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 47, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$x^{2} \\cos{\\left (y \\right )} + y^{3} \\sin^{2}{\\left (x \\right )}$$" ], @@ -1412,7 +1412,7 @@ "x ⋅cos(y) + y ⋅sin (x)" ] }, - "execution_count": 46, + "execution_count": 47, "metadata": {}, "output_type": "execute_result" } @@ -1424,12 +1424,12 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 48, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$2 \\left(- 6 y \\sin^{2}{\\left (x \\right )} + 6 y \\cos^{2}{\\left (x \\right )} - \\cos{\\left (y \\right )}\\right)$$" ], @@ -1438,7 +1438,7 @@ "2⋅⎝- 6⋅y⋅sin (x) + 6⋅y⋅cos (x) - cos(y)⎠" ] }, - "execution_count": 47, + "execution_count": 48, "metadata": {}, "output_type": "execute_result" } @@ -1456,12 +1456,12 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 49, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\frac{\\partial^{3}}{\\partial x^{2}\\partial y} \\left(x^{2} \\cos{\\left (y \\right )} + y^{3} \\sin^{2}{\\left (x \\right )}\\right)$$" ], @@ -1473,7 +1473,7 @@ "∂y ∂x " ] }, - "execution_count": 48, + "execution_count": 49, "metadata": {}, "output_type": "execute_result" } @@ -1491,12 +1491,12 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 50, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAUBAMAAAAJnbK1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdrur781mRIlUMhAi\n3ZlAc+EGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBklEQVQYGWNggAOeBAiT7QCIFvqvbKzykYHh\nEIgDAjZg8iuQVGNgiAFzgMRWEIP9B5BIZ2B7AOKAAIcAiPgJJGYx8B4AUmDA7QCkWH8z8DDMZjgL\nEQKR4UDMf4HhGJCSA+KNwsclgNrfAZn1D8C2Au3jbmC7yBvAwJAKFO0PNnEAUjeBNmzg+M15gIFh\nLpAbv4HbAEi5A9UysCoAWQzrgfgvA/cEIAUUBdoBNBUsCnYukAk0AchfACJrgYZ9BzHAvuPcIM/A\nuAFsG+8HiGgf0DUT9MGOfMGwOf7zG7AwUNtmwY3PEoCcKIg6EInwMc8FhCha6MAkdGAMcEjCOPBQ\nh0uDZJBiCACl+DjKzCgdNQAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$F{\\left (x \\right )}$$" ], @@ -1504,7 +1504,7 @@ "F(x)" ] }, - "execution_count": 49, + "execution_count": 50, "metadata": {}, "output_type": "execute_result" } @@ -1517,12 +1517,12 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 51, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAUBAMAAADiqgm2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHa7q2Yiie9Umd3N\nRDIfxLosAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABE0lEQVQYGWNggINUKCsMTDMqu7iyFTCwGUBF\nWRJADI0ABs4VAQw8YA6Qz3gASAhvBBLrExiigBQEFAGp8wJA4jYDwzaoGAODBAMD11cQ7xUD2wa4\nKM8FhnwwT4yBaQHQRCW3S0DtXAsY6hUgapgDGBjEGaIV9jAwsB5g2P8AIso9gYGhkKFeYBoDA+cG\nhn6goNaZ4wwcFxgYBBgOg5TwfWDYD6LrL4BFGRg+QUWBLgW6joEBZAID3z+QKO8GhvwCIOcPAwPI\ntolMHxgUwbZxAjU92c7AwL6Agfcn8wc2oCQXUKGUyaqAuWBtjE6KWqFAE3gegMyBAISPs2FCQBo1\ndGASzAlQFjgkYaLwUIdLg2WQYggAB5A7C1+LV8QAAAAASUVORK5CYII=\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAACwAAAAVCAYAAAA98QxkAAAABHNCSVQICAgIfAhkiAAAApBJREFUSIntlk9oU0EQxn+poo1V6sHify9WsQcNRSmIigcFUTyJoEEEJUcrSC5iESyI2hQ9ePAPnoIeiqIoKoo9K0RThKJUL1Z60EgLth602kTbw0zM9rmbjXk59oPw3pvvm9nZnZ3dwAy4CQwDDSFibAQmgcT/OK0CzgIZ4CuQ12cGSAHrLT6bgD9AMkSyRdwHcsD8SsRJYByZZT9wHTgPXAWymtQkcCjg1wuMAdEaJNymY3T4hN0qfA9scWhagKfAdsO2FpnIjVBpTsc7YAiY5RIcRZIdAJo8wSLAbOO7S313OPS9yu+zxEkr1xXgzqh9ly3gImSPFoCYJ1kb+tTX1Wwx4DeyGOaKXdKkbJXZqdxFW8CkkneqSLZBk33j0aV1jCP63aHft4E6i75R+VdFg1nSuD57LI7rgIMB2yhwWd+XI6uW8yR8GjgAdCLdfw54BhxG9n8Q34CfyIn1D74js1li4U4qZ/6eGPxmSivlwwUjxgtgnkf/CakeUCrDQsNx2OKUQpojArSrLWvw4/qsryDhEeM9Afzw6KNG/L+YS2nWCzwB0qrba9iWqe25xzeOlD6n+msefZ3qP9jIfuzHThADqlts2CJIZUasHoI9wATSmE3IGZtH+sOFFh3rno1MKPkRaHYEaEWOpiELd1f9bb5bkdIPAkvVtl/1D8okXLwX2l2CKyqYAB4jt14KuAW8Va6AdHcQceWPBewx5LrOAasDXFZ9tjny6dHxVroSBthN6Y9HHjk9BoGHwAlghcNvDvAFeGnYmtU2Cmyw+BQvhoyFa0SarVwFQuOUJtBag1jHKb/6NUE9sr8fhYwTBT4jfTENzn9BVaIAvEaOyT5kS1WDNcAvpIfGapPaDCrDFKa2qh2elt8VAAAAAElFTkSuQmCC\n", "text/latex": [ "$$G{\\left (x \\right )}$$" ], @@ -1530,7 +1530,7 @@ "G(x)" ] }, - "execution_count": 50, + "execution_count": 51, "metadata": {}, "output_type": "execute_result" } @@ -1550,12 +1550,11 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 52, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOUAAAA4CAMAAAA1t8/VAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACXHtMAAAAEnRSTlMAMrvv\nmVREEIndzSJ2Zqvz+70qJfxwAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFuElEQVRoBdVa7barKAxF\nRUFEZob3f9gJhAAC52i70rt6/VPFsGFLCPmoEN9/Td8/RYYZLgwY3w/BzXKaF8nKmgWQm6UQfmVl\nyQLIznLzipclByA7S73zkhQcgOwsDe+2FIIDkJPlIbWdOLclGyAjy205hJgZtyUfIB9LtZywI6Vh\n25aMgHwsZVzFXbOxZATkY7nMwE/5jY0lI+A/XJNSPqzi6rnw4IPxAf7LNSsVjetsxAo2iOPiBOTT\nWGOFOHcn2DYmIyAfS+Wsng5nbzbmESxxdZ0/Lv1DwIBFoOriXRZoPpbVzC+3ys7OuVMccZGPYKMu\nl/uR5kWsfrCz9wF0XjwoEJAk0Ga3ZOiPs7TxGBWrxF/TcVLvnLA+dZqjZhDoNIM+VcuZoT/MUs17\nGlX6wO8ceLk6rke9Vrf3W7S/AS/sDwLVXlp7ASPoD7M02eHbYqyyd0sJZ+zrc9DkLa8ho5NAT1Tf\n+hMR9Osj1Ch399JnWzOFVTyGYZm5MVj9KPjx4IutoCkEagZcEvTgTQ/6bsvhCysVVtEmhVXWumOz\nSaHkQI1/HzLikmEl0LkMlnsn6I+y1K0OobEQCsztuVuxow05X7U/mw/fhSxoAhXTAtsy6s6UARP0\nR1nuvkmE7pgTCnZQw45Nhn8dLEJejdGN9sa5haaeQOE80dbCAJN0GTBBk+gI7GGbH1/Qe0lerZXG\nmHBcLrgDg9k1dMbBtGrndwxGrTiluC2VwwcCnegZdmtmmaD/S6If+dnz9DXGKollGKxS5gvLJxOJ\n21JRrjCB0pEFAB1LhrX8eV54Rob3BvmScoX1K8qMh8zPMO2bfFriCwRdi25ULBP0R1kedHhPyVdJ\nhgLsvw20LXoMr1qfpBhEHkHtkOWfsD4QbkZXZHIphaDjmbHBmT4DSzoK5IthTNkIkSiBFpSisQn6\no2sJfo2cnZSnWNHsoAJBsGGVlSd5DKYoL61P+b2GGYDojPdOVj5U0krtvUxAhWWC5mP5qKAx8vDI\ndynM4C6jNWHGRSg9JNBDL36PPDNLguZj+aigQY51PVlyqes2uEdXtQ0zGqH4WEBX9DNWYkXQ9Dzq\n/WLbo4LGXKka4ufw6DocovVhxlUKnxIo2LITDs1D7t5Fe5ChGVk+KmjkgDfPtucdX0W0QZiR+1U3\nCfQA7xitdnqXoRlZPitokGGlSZa0BbXgb0QbhRlXMXxqQbG1QPOwLAWNa7QxmtF9W0EbhRn3/XsJ\nFpaloNFEG/14D1oK2jDMeIDQiXBknauCRhNtdMM9aKjQhmHGA4hOhCPrXBU0mmijG+5BQ4U2DDMe\nQHQiHBrbFDSqaKMb7kFDhTYMMx5AdCJvsKxOvOhpNAWNOtrohrtvqNDGYcY9RC9BLMmn7CVCy+mc\n1MpGfsUphkAuuKd1QaOJNsZov7ZWaOMw49feP7xMLLf1pKC0FzxM+AZqjrKUbUGx6DWXgkYbbfRY\n9y01Wvmi2TW9BxhIIEtKgw0EwGNKAe8aUg5TGTgIx7REKWi00cYQ76axoEFuaBBm3HQfvo4sVRW4\n91JUXp7CiromSqLkfd+No2UQZrwFG1lOOQM+wLCXl5g30lZONhLG1OCgG1dTG2a8hYsau88aqyjo\n7TpIueEFy5eSpoiOARtkMpwRezA9UY3fGvlRpy7MeNSrE0KWUuqw3zYd/szSXDHDCzUrE/IrW8yu\ngNZGgxS6VOmWpiPHYxdmvAcaWRa7OYjlaVsuIWTbKOkZK1jQUNLY743/Z3oFlkepUyDLorHALP2D\n54gZ1SktXUihRjuUaf+Z6b45SlibyscYrKWAggZc6F6qYH0OH/8CGM+U83qyBMkvvdS+6LQfRyyF\nBsdHY8FcBNcSTrRNp6KvxOTclzK7Tus03sRcwpBlLRvKLfWVq0t149feH6GeOVkvGxbdhK82dQ2e\nwl9znSH59ei68FLFbj3q/PcI1WngUOj4H12NK7blgkUBAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\frac{d}{d x} G{\\left (x \\right )} \\left. \\frac{d}{d \\xi_{1}} F{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=G{\\left (x \\right )} }}$$" ], @@ -1565,7 +1564,7 @@ "dx ⎝dξ₁ ⎠│ξ₁=G(x)" ] }, - "execution_count": 51, + "execution_count": 52, "metadata": {}, "output_type": "execute_result" } @@ -1584,12 +1583,12 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 53, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAUBAMAAAD8YzkFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA70lEQVQYGWNgQAM1Kscc0IQYeAKyuRoYGJWM\nkWVYGPYB1YUxsP9AUf4RyEtlYJiJLMjygYWBYRYDQ74DQlSefQE7A8P+ABTBsBM1pWAl6wNidBgY\npzKw1hwSgOnh+M5QcJOB5wNDlwBjAUyQK4HxgjYDswHjXIYgB5igEgMrwxwG/gNs8848hYnxKDAw\ncHxl8HfgT4AJMTBcYmAUYNrAcJKBH24gA4sCA48AcwKDGQMnUPAARLFUeVEWA5MBxz8Gng0MQQIQ\nwf3///9iYE0/BPTyoXMPIGIwEmgqOshmiG9AF2NYx2CDIcbQc8gBWRAA4JA0cZCNwNMAAAAASUVO\nRK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$2 y e^{x}$$" ], @@ -1598,7 +1597,7 @@ "2⋅y⋅ℯ " ] }, - "execution_count": 52, + "execution_count": 53, "metadata": {}, "output_type": "execute_result" } @@ -1611,12 +1610,12 @@ }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 54, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAALwAAAAZBAMAAAB5tlnFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$4 y^{2} e^{2 x} \\cos{\\left (x \\right )} + 2 y e^{x}$$" ], @@ -1625,7 +1624,7 @@ "4⋅y ⋅ℯ ⋅cos(x) + 2⋅y⋅ℯ " ] }, - "execution_count": 53, + "execution_count": 54, "metadata": {}, "output_type": "execute_result" } @@ -1638,12 +1637,12 @@ }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 55, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$- 4 y^{2} e^{2 x} \\sin{\\left (x \\right )} + 8 y^{2} e^{2 x} \\cos{\\left (x \\right )} + 2 y e^{x}$$" ], @@ -1652,7 +1651,7 @@ "- 4⋅y ⋅ℯ ⋅sin(x) + 8⋅y ⋅ℯ ⋅cos(x) + 2⋅y⋅ℯ " ] }, - "execution_count": 54, + "execution_count": 55, "metadata": {}, "output_type": "execute_result" } @@ -1683,12 +1682,12 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 56, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\frac{x}{2} + \\frac{1}{2} \\sin{\\left (x \\right )} \\cos{\\left (x \\right )}$$" ], @@ -1698,7 +1697,7 @@ "2 2 " ] }, - "execution_count": 55, + "execution_count": 56, "metadata": {}, "output_type": "execute_result" } @@ -1710,12 +1709,12 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 57, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\frac{1}{2} \\log{\\left (\\cos{\\left (x \\right )} - 1 \\right )} - \\frac{1}{2} \\log{\\left (\\cos{\\left (x \\right )} + 1 \\right )}$$" ], @@ -1725,7 +1724,7 @@ " 2 2 " ] }, - "execution_count": 56, + "execution_count": 57, "metadata": {}, "output_type": "execute_result" } @@ -1737,12 +1736,12 @@ }, { "cell_type": "code", - "execution_count": 57, + "execution_count": 58, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAnBAMAAAAY17BRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABzUlEQVRIDe2VPUjDQBzFX2xNm9DWoOLa+LmJ\n4uYWUHESOzmJ7SCCoOjgJkKpH6MUnOJUHXVoHQXFrg5iVdBBCjo5alA3JcZcLr2eXCsuInhDcnm/\ne/e/O7g8QNSaO0d6dBEU6lIqlAlnhVgEZE22VENEhboEpSCEtUCTXosK2XlRiMRA1frQoIm5gCRz\nJxgVsBpyS0fzzWIN/o/+5AnYP2q/uVXxxZRvL+jC5o+5i6FaFH15T2PQ02QjkK3GbWLbGs4NMjhs\nRF6c3tjtETW387YKW8d4wrMlgq+A1I8FT4jscDaGAXl/R597CWiI3ZF5ZInadonAMCBDNOcZ1oFY\nGoE3oizxNoZBLvm2eaenWNQm6bytwoAV3xXyJghbCF5f3qvl8vsigd4inQ+PoTHVWASJhQlMuqOS\nKUwZyhMQ5PbmUMpO9/Y1EguR7b0D17YK6QHRArD8nHMFVKpR9mjbILEQs22nAhAtIWq5x0M8s6a5\naZpb7kcVq46FXuc4C4gXicl9+tU4xsSCWsJwPIU84/IXyTMmFs6ArngOh2yY0WrVjI2F4MbMVVrR\nQ6syU86zcYyNBcX5KaSlgbmh7q82jtWNBbpIZqbvdFvrDfoAzam+cx69TUIAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\frac{x}{2 a^{4} + 2 a^{2} x^{2}}$$" ], @@ -1753,7 +1752,7 @@ "2⋅a + 2⋅a ⋅x " ] }, - "execution_count": 57, + "execution_count": 58, "metadata": {}, "output_type": "execute_result" } @@ -1785,12 +1784,12 @@ }, { "cell_type": "code", - "execution_count": 58, + "execution_count": 59, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAA6BAMAAAB1z6tMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nCoz9138AAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\left(\\frac{x}{\\tan{\\left (x \\right )}}\\right)^{\\frac{1}{x^{2}}}$$" ], @@ -1804,7 +1803,7 @@ "⎝tan(x)⎠ " ] }, - "execution_count": 58, + "execution_count": 59, "metadata": {}, "output_type": "execute_result" } @@ -1824,12 +1823,12 @@ }, { "cell_type": "code", - "execution_count": 59, + "execution_count": 60, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAWBAMAAAC8tehJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjUlEQVQYGWNgQAEbUXhATlIrughDKa1EFn+Z\ngGE0TIDzHRA8ZOD/////B5gYhF5jsgBVgOEZ5wU0EQEmBaAIo7JJAEyCSw/ESg9gVYCJMLABdbF2\nMogsgIkYMHxlYODuMT4IE2Bs4GpgYODfAOMD6UPWC4AiCkgiYCYHUMQARZD5AoMI3HaIjInNARQl\nANmrHvHTrS/pAAAAAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$e^{- \\frac{1}{3}}$$" ], @@ -1838,7 +1837,7 @@ "ℯ " ] }, - "execution_count": 59, + "execution_count": 60, "metadata": {}, "output_type": "execute_result" } @@ -1863,12 +1862,12 @@ }, { "cell_type": "code", - "execution_count": 60, + "execution_count": 61, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAQBAMAAAAG6llRAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\nkgMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABlSURBVAgdY2AAA2Uv6wUQFrPAXvYECJOL4Q6EASKf\nwplcD7ig7DimAiYoU9pMWRHIZFQ2CYCKpAewKkCYrJ0MIgsgTO4e44MQFgPfBiiDgYFPAc7kADIN\nIDzmCwwiMMNMbA5ABAE+RA6ieYoxNgAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$e^{x}$$" ], @@ -1877,7 +1876,7 @@ "ℯ " ] }, - "execution_count": 60, + "execution_count": 61, "metadata": {}, "output_type": "execute_result" } @@ -1890,12 +1889,12 @@ }, { "cell_type": "code", - "execution_count": 61, + "execution_count": 62, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$" ], @@ -1906,7 +1905,7 @@ " 2 6 24 120 " ] }, - "execution_count": 61, + "execution_count": 62, "metadata": {}, "output_type": "execute_result" } @@ -1925,12 +1924,12 @@ }, { "cell_type": "code", - "execution_count": 62, + "execution_count": 63, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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\n", "text/latex": [ "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320} + \\frac{x^{9}}{362880} + \\mathcal{O}\\left(x^{10}\\right)$$" ], @@ -1941,7 +1940,7 @@ " 2 6 24 120 720 5040 40320 362880 " ] }, - "execution_count": 62, + "execution_count": 63, "metadata": {}, "output_type": "execute_result" } @@ -1960,12 +1959,12 @@ }, { "cell_type": "code", - "execution_count": 63, + "execution_count": 64, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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UUh9DGhDrkG3m7kG2CzXhEwdkGsPNiCOtQba1m4h/izrC3Xmq9IM0amQX9i66+fhafH9X33FNmw0R7WXTg4HbgGeRQnItssjtfuCrmLdvjSmqXY38SkLZ9gF3GzXSPmCX7/ZQYZaQ/2bxGiE/pGGjdY1s7Todw6fp3Qq+DdXqXcPfV03XaXaDo2pfta1fuRDRXjrUKMeHdCJaSwffNMXUcCvXGka8Dd+QZt0w0JIEP+h9BJu2PsFGzSNo3VyC3v4E7YSgQy9jAQ3eUivQFgQ/6H0Em7Y+wUbNI2jdXILe/gTthKBDIBAIBAKBQCAQCAQCgUAgEAgEAoFAIBAIBAKBQCAQCAQCgUAgEAgEAoFAoNfxP/FKuIMrj8ZcAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\frac{x^{9}}{362880} + \\frac{x^{8}}{40320} + \\frac{x^{7}}{5040} + \\frac{x^{6}}{720} + \\frac{x^{5}}{120} + \\frac{x^{4}}{24} + \\frac{x^{3}}{6} + \\frac{x^{2}}{2} + x + 1$$" ], @@ -1976,7 +1975,7 @@ "362880 40320 5040 720 120 24 6 2 " ] }, - "execution_count": 63, + "execution_count": 64, "metadata": {}, "output_type": "execute_result" } @@ -1987,12 +1986,12 @@ }, { "cell_type": "code", - "execution_count": 64, + "execution_count": 65, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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\n", "text/latex": [ "$$\\frac{x}{2} - \\frac{1}{10080} \\left(x - \\frac{2 \\pi}{3}\\right)^{7} - \\frac{\\sqrt{3}}{1440} \\left(x - \\frac{2 \\pi}{3}\\right)^{6} + \\frac{1}{240} \\left(x - \\frac{2 \\pi}{3}\\right)^{5} + \\frac{\\sqrt{3}}{48} \\left(x - \\frac{2 \\pi}{3}\\right)^{4} - \\frac{1}{12} \\left(x - \\frac{2 \\pi}{3}\\right)^{3} - \\frac{\\sqrt{3}}{4} \\left(x - \\frac{2 \\pi}{3}\\right)^{2} - \\frac{\\pi}{3} + \\frac{\\sqrt{3}}{2}$$" ], @@ -2012,7 +2011,7 @@ " 4 3 2 " ] }, - "execution_count": 64, + "execution_count": 65, "metadata": {}, "output_type": "execute_result" } @@ -2044,12 +2043,12 @@ }, { "cell_type": "code", - "execution_count": 65, + "execution_count": 66, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAUBAMAAABvx07iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABMUlEQVQoFWNggIDcvQJQFpEUmwPzBCKVQpVx\nOPB8IVFHAMtX0nQwMHB9JFUHhwKpOnKJ0MCSlr0Aroz9ApyJm+HLwPUdLhvGEANn42ToL2D4AZPk\nmbVqI4yNm44LYPwJk+X7//8DjI2PBrtKSNlVTQFTFaOSSVAShrBEAgMDYwJ7EweWFCLGUKHQCdKR\nORME5oI1u/UKMDCwCbB95HIA81GIRIZ8gSUoIiAO21OgHQycG+ASjDM6gKDzAFBAgGE2XBiJ8Q5o\nCQO/ApIIEvMTEhvCdGFg2L8AyDx/AMJHI3n+QAQyQNZ2NIM4/wVAOrgE9BmYQFahgYVMHxgU0cRe\nMzD0OzDEL9jP4I4mA+Sy/uL+wFKAJl7OwAV0qrCS0HV0GaBCRiNF3XI0DQysd28HMAAA7/dBoTLR\np+8AAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$x^{2} - x = 3$$" ], @@ -2058,7 +2057,7 @@ "x - x = 3" ] }, - "execution_count": 65, + "execution_count": 66, "metadata": {}, "output_type": "execute_result" } @@ -2071,12 +2070,12 @@ }, { "cell_type": "code", - "execution_count": 66, + "execution_count": 67, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAAUBAMAAABc0JOuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$x^{2} - x - 3 = 0$$" ], @@ -2085,7 +2084,7 @@ "x - x - 3 = 0" ] }, - "execution_count": 66, + "execution_count": 67, "metadata": {}, "output_type": "execute_result" } @@ -2097,12 +2096,12 @@ }, { "cell_type": "code", - "execution_count": 67, + "execution_count": 68, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAA/BAMAAAACgHlSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+2e2ob8fY1J12/wDSwAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$\\left [ \\frac{1}{2} + \\frac{\\sqrt{13}}{2}, \\quad - \\frac{\\sqrt{13}}{2} + \\frac{1}{2}\\right ]$$" ], @@ -2112,7 +2111,7 @@ "⎣2 2 2 2⎦" ] }, - "execution_count": 67, + "execution_count": 68, "metadata": {}, "output_type": "execute_result" } @@ -2132,12 +2131,12 @@ }, { "cell_type": "code", - "execution_count": 68, + "execution_count": 69, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAATBAMAAAAjVk05AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLUlEQVQoFWNgQAIiJvOQeHiYF6qY8cgiS7kg\nc/CwDZZz4pFFknrp7IPEI5nJ5vXuJaMBsjb+/0CALABhu05g4FWdgCmOLpK1DSiiL4AqzPgAlQ/i\n9ScAiVIQCwlwOiBxIEymvyD6EBAzKquEMTAkuygC2daaIFEUIL8BxE0E4lIBjg8M7BsZhIFsbpAg\nEFh0gEATiGnvACKBgC2SgWkBg+gFNh0gJwMshELsPwDlMn1j4HZgeDrFMwEocAkzZuJB4gxAgnUB\nQ/4Bhn9ADhDIOUJoJHI/iM17gIEh34DhPgPDZyQpINN6NwjsAwmCQ5AJyMi/wLCCVyCYgYEFJIwJ\n5EGRZwbEHA6c+5gZPBjYQMGDBfD+BDo9ASjB9sRGzp2BRWUmFkVgoaR1jwqADADyDDxaVpyiwAAA\nAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$a e^{\\frac{x}{t}} = C$$" ], @@ -2148,7 +2147,7 @@ "a⋅ℯ = C" ] }, - "execution_count": 68, + "execution_count": 69, "metadata": {}, "output_type": "execute_result" } @@ -2162,12 +2161,12 @@ }, { "cell_type": "code", - "execution_count": 69, + "execution_count": 70, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\left [ t \\log{\\left (\\frac{C}{a} \\right )}\\right ]$$" ], @@ -2177,7 +2176,7 @@ "⎣ ⎝a⎠⎦" ] }, - "execution_count": 69, + "execution_count": 70, "metadata": {}, "output_type": "execute_result" } @@ -2212,12 +2211,12 @@ }, { "cell_type": "code", - "execution_count": 70, + "execution_count": 71, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$y{\\left (x \\right )} + \\frac{d}{d x} y{\\left (x \\right )} + \\frac{d^{2}}{d x^{2}} y{\\left (x \\right )} = \\cos{\\left (x \\right )}$$" ], @@ -2229,7 +2228,7 @@ " dx " ] }, - "execution_count": 70, + "execution_count": 71, "metadata": {}, "output_type": "execute_result" } @@ -2243,12 +2242,12 @@ }, { "cell_type": "code", - "execution_count": 71, + "execution_count": 72, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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oitK0TKMyv9o26GoWMs8vzNHIALg+yGo9XzDbx+K2zraiKIqiKIaxyNuTtgNOidi/MzLy/D3kFWLBZz3SWlYURVGU1EQtuN3MzELmXC8lenL3MGR5u5ER+zbkaJeiKIqyGaPOuJp3kTeSjCB6YYyNyKpCq5C1chVFURSlbnQ0dUceAl6z7JuH9A3fjTjsXYGjkMXhfa8lrSiKoihNywCqpynVMhJ57/BapHW8CAltK4qiKIqiKIqiKIqiKIqiKIqiKIqiKIqiKIqiKIqiKIrSlfg/mtETl2KN/mcAAAAASUVORK5CYII=\n", "text/latex": [ "$$y{\\left (x \\right )} = \\frac{1}{\\sqrt{e^{x}}} \\left(C_{1} \\sin{\\left (\\frac{\\sqrt{3} x}{2} \\right )} + C_{2} \\cos{\\left (\\frac{\\sqrt{3} x}{2} \\right )}\\right) + \\sin{\\left (x \\right )}$$" ], @@ -2262,7 +2261,7 @@ " ╲╱ ℯ " ] }, - "execution_count": 71, + "execution_count": 72, "metadata": {}, "output_type": "execute_result" } @@ -2281,12 +2280,11 @@ }, { "cell_type": "code", - "execution_count": 72, + "execution_count": 73, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAyBAMAAAAKOF7GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhAimXZEzWa7\n3asFX9bSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABSklEQVQ4Ee3UPUsDMRwG8Kdpc6WnxYJ2suIN\nbiLcN6hg3c/JRbj7BFKKgzhoh0J1s59AFyk4iKNjnQQnwd3FpeAiSgvS4UzIi3/hLrODWS7J75K7\nJA/BavqBrMLStIbF1naWgbe2aljKJNlZysa9FQfixoWfDqxMHMieCHqdw7psmuKP+6H920HivxuQ\nz/iKTw3yZZRHFJsRxLapdZYnqHYpHoFbLIwQb1IcoiIWo0bGAZrUMIOYzGCIx2JC+As7twb97twL\nI4Y3PIimmpaP1/bPKO72xUCNtJ/U1UjSQav/KHYjdxNK9+18xELowIvIgRuOb2L4g+y0Rw/L6/Rs\nwOA945riIJm3AUMhwCtBGfJLO+1JRAgyl9XAYuOXQYb83KDMPi065Hr7ZkCRaKxCrvEY/ICgDrlG\ntn5HDDrkuaci3/2L6LxSXZfxN0SVSvCGgXwsAAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$$" ], @@ -2296,7 +2294,7 @@ "⎣c d⎦" ] }, - "execution_count": 72, + "execution_count": 73, "metadata": {}, "output_type": "execute_result" } @@ -2313,12 +2311,12 @@ }, { "cell_type": "code", - "execution_count": 73, + "execution_count": 74, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": "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\n", "text/latex": [ "$$\\left \\{ \\frac{a}{2} + \\frac{d}{2} - \\frac{1}{2} \\sqrt{a^{2} - 2 a d + 4 b c + d^{2}} : 1, \\quad \\frac{a}{2} + \\frac{d}{2} + \\frac{1}{2} \\sqrt{a^{2} - 2 a d + 4 b c + d^{2}} : 1\\right \\}$$" ], @@ -2336,7 +2334,7 @@ " ⎭" ] }, - "execution_count": 73, + "execution_count": 74, "metadata": {}, "output_type": "execute_result" } @@ -2348,30 +2346,25 @@ }, { "cell_type": "code", - "execution_count": 74, + "execution_count": 75, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAABLCAMAAAB5hvoWAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n6NG26sZzqprPqQ8LIZnEX1RvU+sec8S+AAAAAElFTkSuQmCC\n", "text/latex": [ - "$$\\left[\\begin{matrix}\\frac{1}{a} + \\frac{b c}{a^{2} \\left(d - \\frac{b c}{a}\\right)} & - \\frac{b}{a \\left(d - \\frac{b c}{a}\\right)}\\\\- \\frac{c}{a \\left(d - \\frac{b c}{a}\\right)} & \\frac{1}{d - \\frac{b c}{a}}\\end{matrix}\\right]$$" + "$$\\left[\\begin{matrix}\\frac{d}{a d - b c} & - \\frac{b}{a d - b c}\\\\- \\frac{c}{a d - b c} & \\frac{a}{a d - b c}\\end{matrix}\\right]$$" ], "text/plain": [ - "⎡1 b⋅c -b ⎤\n", - "⎢─ + ──────────── ───────────⎥\n", - "⎢a 2 ⎛ b⋅c⎞ ⎛ b⋅c⎞⎥\n", - "⎢ a ⋅⎜d - ───⎟ a⋅⎜d - ───⎟⎥\n", - "⎢ ⎝ a ⎠ ⎝ a ⎠⎥\n", - "⎢ ⎥\n", - "⎢ -c 1 ⎥\n", - "⎢ ─────────── ─────── ⎥\n", - "⎢ ⎛ b⋅c⎞ b⋅c ⎥\n", - "⎢ a⋅⎜d - ───⎟ d - ─── ⎥\n", - "⎣ ⎝ a ⎠ a ⎦" + "⎡ d -b ⎤\n", + "⎢───────── ─────────⎥\n", + "⎢a⋅d - b⋅c a⋅d - b⋅c⎥\n", + "⎢ ⎥\n", + "⎢ -c a ⎥\n", + "⎢───────── ─────────⎥\n", + "⎣a⋅d - b⋅c a⋅d - b⋅c⎦" ] }, - "execution_count": 74, + "execution_count": 75, "metadata": {}, "output_type": "execute_result" } @@ -2383,12 +2376,11 @@ }, { "cell_type": "code", - "execution_count": 75, + "execution_count": 76, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKMAAAAzBAMAAAAJJKXwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left[\\begin{matrix}a^{2} + b c & a b + b d\\\\a c + c d & b c + d^{2}\\end{matrix}\\right]$$" ], @@ -2400,7 +2392,7 @@ "⎣a⋅c + c⋅d b⋅c + d ⎦" ] }, - "execution_count": 75, + "execution_count": 76, "metadata": {}, "output_type": "execute_result" } @@ -2473,7 +2465,7 @@ }, { "cell_type": "code", - "execution_count": 76, + "execution_count": 77, "metadata": {}, "outputs": [ { @@ -2498,7 +2490,7 @@ }, { "cell_type": "code", - "execution_count": 77, + "execution_count": 78, "metadata": {}, "outputs": [ { @@ -2670,7 +2662,7 @@ "" ] }, - "execution_count": 77, + "execution_count": 78, "metadata": {}, "output_type": "execute_result" } @@ -2699,7 +2691,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.4" + "version": "3.6.4" } }, "nbformat": 4, diff --git a/notebooks_completos/041-SymPy-Mecanica.ipynb b/notebooks_completos/041-SymPy-Mecanica.ipynb index 29a996e..ceba149 100644 --- a/notebooks_completos/041-SymPy-Mecanica.ipynb +++ b/notebooks_completos/041-SymPy-Mecanica.ipynb @@ -37,32 +37,35 @@ "El objeto primordial que vamos a manejar van a ser los sistemas de referencia. Podremos definir relaciones geométricas entre ellos y de esta forma las transformaciones de vectores entre un sistema y otro serán triviales." ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "La manera usual de empezar a trabajar con SymPy es importar la función `init_session`:\n", + "```\n", + "from sympy import init_session\n", + "init_session(use_latex=True)```\n", + "\n", + " Esta función ya se encarga de importar todas las funciones básicas y preparar las salidas gráficas. Sin embargo, en este momento, esta función se encuentra en mantenimiento para su uso dentro de los notebooks por lo que activaremos la salida gráfica e importaremos las funciones de la manera usual. Puedes consultar el estado de la corrección en: https://github.com/sympy/sympy/pull/13300 y https://github.com/sympy/sympy/issues/13319 ." + ] + }, { "cell_type": "code", "execution_count": 1, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "IPython console for SymPy 1.0 (Python 3.5.4-64-bit) (ground types: python)\n", - "\n", - "These commands were executed:\n", - ">>> from __future__ import division\n", - ">>> from sympy import *\n", - ">>> x, y, z, t = symbols('x y z t')\n", - ">>> k, m, n = symbols('k m n', integer=True)\n", - ">>> f, g, h = symbols('f g h', cls=Function)\n", - ">>> init_printing()\n", - "\n", - "Documentation can be found at http://docs.sympy.org/1.0/\n" - ] - } - ], + "outputs": [], "source": [ - "from sympy import init_session\n", - "init_session(use_latex=True)" + "from sympy import init_printing\n", + "init_printing()" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy import symbols" ] }, { @@ -76,7 +79,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": {}, "outputs": [], "source": [ @@ -85,12 +88,12 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 4, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAUBAMAAAB7UtADAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAECLdzUTvdomZVDK7\nq2ayhyTRAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiUlEQVQIHWNgYGBkQABlRzibNcVMAcaRL+BI\ngLGBygVgbGS6xcVSAcrn+sawPwnG/szA8xOmbiMD5xcYW6rnMozN9b8RLq7/D6Rm8X+BfEOG+E8M\nPEA154NSGRj4vzPs+azAwPxtAQMD07UzDee/MzB9QziE574BzGCGu6x/YOxn3xn+Z8E4IBoAX5om\nBf/HSGkAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\mathbf{\\hat{a}_x}$$" ], @@ -98,7 +101,7 @@ "a_x" ] }, - "execution_count": 3, + "execution_count": 4, "metadata": {}, "output_type": "execute_result" } @@ -117,12 +120,12 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 5, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAWBAMAAABzm9yKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFW3RMUvDQBQH8P+RpKmhxrqIurT2\nCygIdUy+gTe5uHTRwUVxEbeuTirSQUFaVx1S0KkOzeQsLo7xKwhKkyicZxJq3nlvuXs//u/uSIC8\neLGqi+rsTpV8ouSste4DJ6Nz9aysL/kW7BjWN451wbLvAJcAh6ULlv0K2Pe1IQXHXAbtx9GK4lmr\neMCjMzOpa5LUqxNEA3hNXZC404HJsTrUBKm3AOu1HfwFt29/60EOEq/JO6MJ155I/AVsXrzJq2eD\nz5mU/EmWudvvmUkXZhO1RVGHN4QRL3XJQ6u5NwboActHh3sQoendAME1yaHwSuzK542FSLFwsWGI\nEA05SCp388tRPnI7JbFp4+1Ot9mm0jldo1J0m++E3X44lzwTKhpDP69GWej4qml7++NJ6/+Q3R9I\n+wHISmMT/TXw5wAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$2\\mathbf{\\hat{a}_x} - \\mathbf{\\hat{a}_y}$$" ], @@ -130,7 +133,7 @@ "2 a_x - a_y" ] }, - "execution_count": 4, + "execution_count": 5, "metadata": {}, "output_type": "execute_result" } @@ -151,12 +154,12 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 6, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAATBAMAAADMlCYHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu91EzRCrMiKZVHaJ\n72YDRvcCAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVQoFWNgEDJgQAG9qihc9v+/wXzm/w4o\n4gzz/6HyayHqWP8noIq//47KZ4GoYyiGCq+G0lxQd3BtgAjA1MF0w9TB+Gjq+GH2QtUx//8DUYmk\nbj1QDYd/AkQcZl4TprreVKCS+WjqGDHUsd8BmURYnf0DuLojLi66Li5uID7YPB4XF6crLi4JQD7L\nl/8girB5n96DQ56gvb+4/zcQY94vBnkF4tT1/2Rg4P//H6gYCGDhhxYu7P//O5z//wOiBqGOtQBN\nHUIBlBUBpvm/8/6CCLA2QCWwUvyf6xWwSqAJMjtpBqAJgbkA41Varlo3R+MAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\mathbf{i} + \\mathbf{j} + \\mathbf{k}$$" ], @@ -164,7 +167,7 @@ "1_x + 1_y + 1_z" ] }, - "execution_count": 5, + "execution_count": 6, "metadata": {}, "output_type": "execute_result" } @@ -183,7 +186,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 7, "metadata": {}, "outputs": [], "source": [ @@ -199,12 +202,12 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 8, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGcAAAATBAMAAAB/3A4dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu91EzRCrMiKZVO92\niWZOMaHgAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABK0lEQVQoFWNgEDJgwAAcmwIwxNj//4GLMf93\ngLPhDM7/D+BsOKMaoYn1fwJcGM7g+18AZ8MZLAhNDMVwUSBjDZSzGIsgsiZkabgmbIJImvhRnAe1\nyf7/BSRtUEGgpvVQxRz+CZjyHPrYNXWnQtXOB2oqgemD+Un+AgO7A5ogyx/2vUAhsDhQU2USmjyD\n/AX2aQpogix/7B8A9YDFQTYtAstfcXHRdnFxA7GBNnEpoAmyfAH7CCyO0ARUhOQ8qCaEIMun9/+A\nPBI1/eL530C6JgZ5BTI09f+EaOL//x8WEAjnIwUEXJD9/3+H+/9/wAIIQ1M4Nk1A3WAAD6DVMJEI\nMON/wf4FDFwbUAVhPJh4p34OTAiiyfwTA8ekbwdQBKEcXOIMJkpTsamHiwEA3YRstqP1a+gAAAAA\nSUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\mathbf{i}_{1} + \\mathbf{j}_{1} + \\mathbf{k}_{1}$$" ], @@ -212,7 +215,7 @@ "1_x + 1_y + 1_z" ] }, - "execution_count": 7, + "execution_count": 8, "metadata": {}, "output_type": "execute_result" } @@ -238,7 +241,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 9, "metadata": {}, "outputs": [], "source": [ @@ -249,12 +252,12 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 10, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAAATBAMAAABvkIKEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "image/png": "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\n", "text/latex": [ "$$R\\mathbf{i}_{1} + R\\mathbf{j}_{1} + R\\mathbf{k}_{1}$$" ], @@ -262,7 +265,7 @@ "R 1_x + R 1_y + R 1_z" ] }, - "execution_count": 9, + "execution_count": 10, "metadata": {}, "output_type": "execute_result" } @@ -273,20 +276,20 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 11, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAASBAMAAABFi4Y1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4RVtAAAAAElFTkSuQmCC\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAXCAYAAAAIqmGLAAAABHNCSVQICAgIfAhkiAAAAvRJREFUaIHt2F+IVFUcB/BP6hK6i1ChRCgsJNIGUeK7joWCkE+WKMY2UA9BEj0sBIkgEoSiRUXgk2uiL5H4YElQD6JCL0JFsIUYmYkPBbmBqevuUg/nbHu9c+/MvevMzkzMFy733vP7c37zPef3O7879PC/RxX/JK7BdgbTAahqAR8LEs87ouOPC9gdjroHmhFEi/EIXsUpXMZt/IULeMW9HMygY7l4Mk52roHeGkzjGp7GSOJa2soA54jXhN91HSfwLo5gPI5/hgdSNnPhYsA8VJaFwi670UDvQgxgW7MDaBGexRa1GfMorgq/ZWtKNlcuquah/F+MzlfkyF+K8q/nM6gW4m0h7o8yZGW5oD4fOzGVkH2KviJBpnfX9/H+VIbuAPZjEruKOO8CTMb7VIasmVzsxCdChsKocO5N5lokUGaR9uAxvIefijjvcCzCcHz+MkPeLC7SC/Sh0LBMlwk2iYqQisdS46sxgd/Qnxiv6t5yd1CI+YsceUU5LqjlY7d7S9w79x82D0Vn36bGz8TxFxoENZiQrcNpoavKsq2HKym/ja7jJXzDG9HuRzyco1OWC2r5SF4jOfM05GlR6v2GsEOGomxK6Iw24yuhXS2KfqFkjOJkCTv4GXdK6F8vofs6PsAYnsOfOXrN5GIav+TI5sTTaWFFh/Cg8AE4IaR5GlXFyl3ZTGoV3hRi+QHLC+iX4YJaPsYSz3fxfIP5MnnK+tpOHpgjeByHcKnBBJ2Ot/A+vsMG/F7A5n652G62XPYJ2bexoG1dvCis6Chu4lcsydGt6o5M2hNjuCj/DMpCGS7I5mOZ0AHOjN3C+hz7QmcSs7unGu/D0XG34mXsE86F80LTkMYVHM0YbwYXf2CT8O/ESizG53Hsm5K+/sMC/C2satb3QxJVnZ9JezXuDs/m2Jbhgvp8PCEs2IxsHGtT9m2tOO0ud92CwuWuWRjAqsT7IJ4RWt6rLZy329BWniqyS8vRVk/cZajo8dRDDz300EOX4F/etym+OEVKOAAAAABJRU5ErkJggg==\n", "text/latex": [ "$$V\\mathbf{i}_{1} - 2 V\\mathbf{k}_{1}$$" ], "text/plain": [ - "V 1_x - 2⋅V 1_z" + "V 1_x + -2⋅V 1_z" ] }, - "execution_count": 10, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } @@ -297,7 +300,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 12, "metadata": {}, "outputs": [], "source": [ @@ -306,12 +309,12 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 13, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAOBAMAAABN+VgMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\niUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyUlEQVQYGWNggAHW/G/p5QEMk/9rMzA4/zoA\nEwbS/AUMHN8YGD8BmdwJSOIM/hMYGPIdGFYBxdiQxRnmOzAw2Dcw6AMFp6BIVAGN+A7U1MDAcQFF\n4hsDw64EBob7D9BMYvk2s/4IUGX8BDSTOD8ysCoCJd4fgJgkZAwCKgwM7AsYGP4CJeQLUN3EwJ8A\n8gbQOwZxQBIJ+AcwcP8B8pl/HEASBTKB3uD9ySDAwP5dAFUC6DHWnywbGHj+oYof+p/KwHB8JlA2\nASwBADazL5flHdcjAAAAAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$- R V$$" ], @@ -319,7 +322,7 @@ "-R⋅V" ] }, - "execution_count": 12, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } @@ -332,20 +335,20 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 14, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAAATBAMAAACO+5fVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "image/png": "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\n", "text/latex": [ "$$- 2 R V\\mathbf{i}_{1} + 3 R V\\mathbf{j}_{1} - R V\\mathbf{k}_{1}$$" ], "text/plain": [ - "- 2⋅R⋅V 1_x + 3⋅R⋅V 1_y - R⋅V 1_z" + "-2⋅R⋅V 1_x + 3⋅R⋅V 1_y + -R⋅V 1_z" ] }, - "execution_count": 13, + "execution_count": 14, "metadata": {}, "output_type": "execute_result" } @@ -365,12 +368,12 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 15, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAVBAMAAADiLT7tAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n82gOg8SBsAAAAABJRU5ErkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$\\sqrt{14} R V$$" ], @@ -378,7 +381,7 @@ "√14⋅R⋅V" ] }, - "execution_count": 14, + "execution_count": 15, "metadata": {}, "output_type": "execute_result" } @@ -389,26 +392,22 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 16, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": "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\n", "text/latex": [ "$$- \\frac{\\sqrt{14}}{7}\\mathbf{i}_{1} + \\frac{3 \\sqrt{14}}{14}\\mathbf{j}_{1} - \\frac{\\sqrt{14}}{14}\\mathbf{k}_{1}$$" ], "text/plain": [ - "- √14 \n", - "─────\n", - " 7 1_x + 3⋅√14\n", - "─────\n", - " 14 1_y - √14 \n", - "─────\n", - " 14 1_z" + "-√14 3⋅√14 -√14\n", + "───── 1_x + ───── 1_y + ───── 1_z\n", + " 7 14 14" ] }, - "execution_count": 15, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } @@ -432,7 +431,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 17, "metadata": {}, "outputs": [], "source": [ @@ -442,12 +441,12 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 18, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAASBAMAAABRHhjUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqu7IjJ23e9UmRDN\niUTiMbt4AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVQYGU2OMW7CQBBFnyDGBANrUSfKioIa\nhES4BWlpyEp0aXKDKMoJEBeAHiF8BG5gKlp8BAKYIhTLLmQxU828+TP/g6vXxHU1XW+3qOqlA+yh\nJwMtHfCOkEsZuJlCCv7hNkL+GyZNkZ0IyeMPD7F0ovfG8O8NFpLCemXhZ8RTdAHPjC3ogjAbo/hC\nmA07G8OCIKU4Au/XmEwt8A74IZcYGxVYcMJXUN7Ci+pfFUVFKd6FlDpKaG1+iIT7+nfJ0Mc1RwYq\n8xlnXb01JCaHWEQAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$R\\mathbf{i}_{0}$$" ], @@ -455,7 +454,7 @@ "R 0_x" ] }, - "execution_count": 17, + "execution_count": 18, "metadata": {}, "output_type": "execute_result" } @@ -467,12 +466,12 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 19, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAASBAMAAADBDWEbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmaJu6sQ\n791S4TDfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVQYGVXOMU4CQRTG8f+wrMuKIVvZGTUe\nQCsTKrkBVNpyBC9glniBNYRQ2HgEPQAFN9BoZWMgNLawQlYljt9IAuMrZl5+efPNA8zRIUQ2Z13l\n82T3Aq48yjpwdkC4oWiu+XDq09anKM59CmeiypejY/ugHrYLHcG3o73BnxAsdFcLUfSqrl6HnTc1\ntSdRs6OEm5IeZwmkWmLukmoNllrgBG51/WRKfbzkGWNaMDaEs8AmpAnvxAww7VjE/Yh0yITqdY9S\n99TR/sJNvRDbgorNI2tbd3aqrA9t4NfqR18I2m6v/9XvD38B5S9IY52D59QAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\Omega\\mathbf{k}_{0}$$" ], @@ -480,7 +479,7 @@ "Ω 0_z" ] }, - "execution_count": 18, + "execution_count": 19, "metadata": {}, "output_type": "execute_result" } @@ -492,12 +491,12 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 20, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmaJu6sQ\n791S4TDfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABNElEQVQYGVWQsUvDUBDGv5dok7TSZnKO6CoU\nCkKnuoijTroIdhFBHPIfNPgP9CHSwclZkHQUEZrNUdG9Ii5OFo1KbYrPL+8JkoPcfe939+7uBYBY\nXMC/iWFiDjNb/vw2pTvIrs4jlFRkeJexFVDXziAyuCrU3PlkmH2j69SBQYgljVH6ZvRSuhYLR76h\nLH2ncid0F4A1wYaSOlUeM9gZHb/lPsTIcPuLqMKkkzXifepYotkE5obU1TvemcLd0dyVVh/o+lwl\nAMrM5TvEsrqKKQfXgVOea+ydj4nlbYh7CLEJPAneiWDls2LZ8fECD9cQbU+vz2ckOU/wjMrRCazj\nFWCXz0idwNQ/wFNj/pMUe+oSOGj89f9gw4LdsN7sU8BQ4WNgt/P9i6bWf4BeLylSYO31UKNfnF5f\nvJfxI8sAAAAASUVORK5CYII=\n", + "image/png": "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\n", "text/latex": [ "$$\\Omega R\\mathbf{j}_{0}$$" ], @@ -505,7 +504,7 @@ "Ω⋅R 0_y" ] }, - "execution_count": 19, + "execution_count": 20, "metadata": {}, "output_type": "execute_result" } @@ -517,12 +516,12 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 21, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAPBAMAAADuVgmRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIlR2RM2ZMmaJu6sQ\n791S4TDfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0klEQVQIHU2OsUrDUBSGvxsbkzRiMnWOOBcC\nga55BSddBF1E6NQ3aPAFAuLk5FwQM5ZSSB+hT1DoE4gGJRoh/snUC/c/H/89554fMOdnHJ7BZTi6\nkuGWzWqRCXJJGgmCF0wDzpfY/pDMYyhnHP+IvUqSzuA9xP4Uu7+SV7BUh7X4SLPojgvxt9iX6zTJ\n2734ZCc53erlD/daTB5qQaRZmd0C7BieVYOij4ExF7A36sqwuv881phbr4+hOBv8hyesxwncKE7l\nRHhtjdtW3LVLmCb8A/w5NMBDzH8OAAAAAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$\\Omega R$$" ], @@ -530,7 +529,7 @@ "Ω⋅R" ] }, - "execution_count": 20, + "execution_count": 21, "metadata": {}, "output_type": "execute_result" } @@ -563,7 +562,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 22, "metadata": {}, "outputs": [], "source": [ @@ -573,12 +572,11 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 23, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMwAAABLCAMAAAALSZigAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )} & \\sin{\\left (\\phi \\right )} & 0\\\\- \\sin{\\left (\\phi \\right )} & \\cos{\\left (\\phi \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" ], @@ -590,7 +588,7 @@ "⎣ 0 0 1⎦" ] }, - "execution_count": 22, + "execution_count": 23, "metadata": {}, "output_type": "execute_result" } @@ -630,12 +628,12 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 24, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAUBAMAAAA0IxGWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "image/png": "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\n", "text/latex": [ "$$\\operatorname{cos}\\left(\\phi\\right)\\mathbf{i}_{1} + \\operatorname{sin}\\left(\\phi\\right)\\mathbf{j}_{1}$$" ], @@ -643,7 +641,7 @@ "cos(φ) 1_x + sin(φ) 1_y" ] }, - "execution_count": 23, + "execution_count": 24, "metadata": {}, "output_type": "execute_result" } @@ -654,12 +652,11 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 25, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAABLCAMAAADtTJGhAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )}\\\\\\sin{\\left (\\phi \\right )}\\\\0\\end{matrix}\\right]$$" ], @@ -671,7 +668,7 @@ "⎣ 0 ⎦" ] }, - "execution_count": 24, + "execution_count": 25, "metadata": {}, "output_type": "execute_result" } @@ -691,7 +688,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 26, "metadata": {}, "outputs": [], "source": [ @@ -700,12 +697,12 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 27, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAUBAMAAAD1iJl/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXaZRCLdEO9Uu81m\nqzIdlvb2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVQYGWNggAHWAyDWURgXREeBObwBSGJb\nGRjYUxnYNyCEeAwYGDg/MDAIIYTYgDqYgUoqEEJhQCabAAMDtwOQIaQMIpUZGHzXr1VgYDZgYAhz\nYG9lncCwBig8H4h5gJobGRgWMxUwZAK51kDMmcDA/B3IZGRgaAFyE4GY7wMDxwIGBglxsBD7F4hQ\nPdAW+QkMII0sHxgKGFgSGPgPAIWAskDjGTdwPgAZz3yBgceer4BBGOjSC0VAAqhL8MwBZqAv3BgY\nWNcAdXA/AGqBAJCHQADkCygAeRsEkLzNsBQsghw40CDkguoHy8MDGgDDjSpJfBFnNAAAAABJRU5E\nrkJggg==\n", + "image/png": "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\n", "text/latex": [ "$$\\alpha{\\left (t \\right )}$$" ], @@ -713,7 +710,7 @@ "α(t)" ] }, - "execution_count": 26, + "execution_count": 27, "metadata": {}, "output_type": "execute_result" } @@ -732,12 +729,12 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 28, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAArBAMAAAAu4Q2VAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWAYkAAAAAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$\\frac{d}{d t} \\alpha{\\left (t \\right )}$$" ], @@ -747,7 +744,7 @@ "dt " ] }, - "execution_count": 27, + "execution_count": 28, "metadata": {}, "output_type": "execute_result" } @@ -771,7 +768,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 29, "metadata": {}, "outputs": [], "source": [ @@ -780,12 +777,11 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 30, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQAAAABLCAMAAACoRKlPAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nK/4DNGVhyUtpMY8AAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\psi{\\left (t \\right )} \\right )} & \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\psi{\\left (t \\right )} \\right )} & \\cos{\\left (\\psi{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" ], @@ -797,7 +793,7 @@ "⎣ 0 0 1⎦" ] }, - "execution_count": 29, + "execution_count": 30, "metadata": {}, "output_type": "execute_result" } @@ -811,12 +807,11 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 31, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP0AAABLCAMAAACMeHGmAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\sin{\\left (\\beta{\\left (t \\right )} \\right )}\\\\0 & 1 & 0\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" ], @@ -828,7 +823,7 @@ "⎣-sin(β(t)) 0 cos(β(t))⎦" ] }, - "execution_count": 30, + "execution_count": 31, "metadata": {}, "output_type": "execute_result" } @@ -842,12 +837,11 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 32, "metadata": {}, "outputs": [ { "data": { - "image/png": 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"text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" ], @@ -859,7 +853,7 @@ "⎣ -sin(β(t)) 0 cos(β(t)) ⎦" ] }, - "execution_count": 31, + "execution_count": 32, "metadata": {}, "output_type": "execute_result" } @@ -873,12 +867,11 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 33, "metadata": {}, "outputs": [ { "data": { - "image/png": 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"text/latex": [ "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\0 & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" ], @@ -890,7 +883,7 @@ "⎣0 -sin(θ(t)) cos(θ(t))⎦" ] }, - "execution_count": 32, + "execution_count": 33, "metadata": {}, "output_type": "execute_result" } @@ -904,12 +897,11 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 34, "metadata": {}, "outputs": [ { "data": { - "image/png": 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"text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & 0\\\\- \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )}\\\\\\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" ], @@ -921,7 +913,7 @@ "⎣sin(θ(t))⋅sin(ζ(t)) -sin(θ(t))⋅cos(ζ(t)) cos(θ(t))⎦" ] }, - "execution_count": 33, + "execution_count": 34, "metadata": {}, "output_type": "execute_result" } @@ -932,12 +924,11 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 35, "metadata": {}, "outputs": [ { "data": { - "image/png": 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"text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} - \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & - \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )}\\\\- \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )} + \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} \\cos{\\left (\\beta{\\left (t \\right )} \\right )} & - \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\cos{\\left (\\zeta\\left(t\\right) \\right )} & \\sin{\\left (\\beta{\\left (t \\right )} \\right )} \\sin{\\left (\\theta{\\left (t \\right )} \\right )} \\sin{\\left (\\zeta\\left(t\\right) \\right )} + \\cos{\\left (\\beta{\\left (t \\right )} \\right )} \\cos{\\left (\\theta{\\left (t \\right )} \\right )}\\end{matrix}\\right]$$" ], @@ -955,7 +946,7 @@ "in(β(t))⋅sin(θ(t))⋅sin(ζ(t)) + cos(β(t))⋅cos(θ(t)) ⎦" ] }, - "execution_count": 34, + "execution_count": 35, "metadata": {}, "output_type": "execute_result" } @@ -975,24 +966,22 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 36, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2}$$" ], "text/plain": [ - "d \n", - "──(θ(t))\n", - "dt a3_x + d \n", - "──(ζ(t))\n", - "dt a2_z" + "d d\n", + "──(θ(t)) a3_x + ──(ζ(t)) a2_z\n", + "dt dt" ] }, - "execution_count": 35, + "execution_count": 36, "metadata": {}, "output_type": "execute_result" } @@ -1003,28 +992,22 @@ }, { "cell_type": "code", - "execution_count": 36, + "execution_count": 37, "metadata": {}, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$\\dot{\\theta}\\mathbf{i}_{A3} + \\dot{\\zeta\\left(t\\right)}\\mathbf{k}_{A2} - \\dot{\\beta}\\mathbf{j}_{A1} + \\dot{\\psi}\\mathbf{k}_{A}$$" ], "text/plain": [ - "d \n", - "──(θ(t))\n", - "dt a3_x + d \n", - "──(ζ(t))\n", - "dt a2_z + d \n", - "-──(β(t))\n", - " dt a1_y + d \n", - "──(ψ(t))\n", - "dt a_z" + "d d d d\n", + "──(θ(t)) a3_x + ──(ζ(t)) a2_z + -──(β(t)) a1_y + ──(ψ(t)) a_z\n", + "dt dt dt dt" ] }, - "execution_count": 36, + "execution_count": 37, "metadata": {}, "output_type": "execute_result" } @@ -1035,26 +1018,35 @@ }, { "cell_type": "code", - "execution_count": 37, - "metadata": {}, + "execution_count": 38, + "metadata": { + "collapsed": true + }, "outputs": [ + { + "ename": "AttributeError", + "evalue": "'tuple' object has no attribute 'right'", + "output_type": "error", + "traceback": [ + "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[1;31mAttributeError\u001b[0m Traceback (most recent call last)", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\IPython\\core\\formatters.py\u001b[0m in \u001b[0;36m__call__\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 700\u001b[0m \u001b[0mtype_pprinters\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_printers\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 701\u001b[0m deferred_pprinters=self.deferred_printers)\n\u001b[1;32m--> 702\u001b[1;33m \u001b[0mprinter\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 703\u001b[0m \u001b[0mprinter\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mflush\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 704\u001b[0m \u001b[1;32mreturn\u001b[0m \u001b[0mstream\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mgetvalue\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\IPython\\lib\\pretty.py\u001b[0m in \u001b[0;36mpretty\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 378\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0mcls\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_pprinters\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 379\u001b[0m \u001b[1;31m# printer registered in self.type_pprinters\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 380\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_pprinters\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mcls\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcycle\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 381\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 382\u001b[0m \u001b[1;31m# deferred printer\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\interactive\\printing.py\u001b[0m in \u001b[0;36m_print_plain\u001b[1;34m(arg, p, cycle)\u001b[0m\n\u001b[0;32m 66\u001b[0m \u001b[1;34m\"\"\"caller for pretty, for use in IPython 0.11\"\"\"\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 67\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0m_can_print_latex\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 68\u001b[1;33m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtext\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mstringify_func\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\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 69\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 70\u001b[0m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtext\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mIPython\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlib\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\printing\\pretty\\pretty.py\u001b[0m in \u001b[0;36mpretty\u001b[1;34m(expr, **settings)\u001b[0m\n\u001b[0;32m 2162\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 2163\u001b[0m \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2164\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mpp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdoprint\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 2165\u001b[0m \u001b[1;32mfinally\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 2166\u001b[0m \u001b[0mpretty_use_unicode\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0muflag\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\printing\\pretty\\pretty.py\u001b[0m in \u001b[0;36mdoprint\u001b[1;34m(self, expr)\u001b[0m\n\u001b[0;32m 60\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 61\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mdoprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\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---> 62\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_print\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrender\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m**\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_settings\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 63\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 64\u001b[0m \u001b[1;31m# empty op so _print(stringPict) returns the same\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\physics\\vector\\vector.py\u001b[0m in \u001b[0;36mrender\u001b[1;34m(self, *args, **kwargs)\u001b[0m\n\u001b[0;32m 283\u001b[0m pform = vp._print(\n\u001b[0;32m 284\u001b[0m ar[i][0][j])\n\u001b[1;32m--> 285\u001b[1;33m pform = prettyForm(*pform.right(\" \",\n\u001b[0m\u001b[0;32m 286\u001b[0m ar[i][1].pretty_vecs[j]))\n\u001b[0;32m 287\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;31mAttributeError\u001b[0m: 'tuple' object has no attribute 'right'" + ] + }, { "data": { - "image/png": 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+ "image/png": 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\n", "text/latex": [ "$$(\\left(- \\operatorname{sin}\\left(\\psi\\right) \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\psi\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\psi\\right) \\dot{\\zeta\\left(t\\right)} + \\operatorname{sin}\\left(\\psi\\right) \\dot{\\beta})\\mathbf{i}_{A} + (\\left(\\operatorname{sin}\\left(\\psi\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) + \\operatorname{sin}\\left(\\zeta\\left(t\\right)\\right) \\operatorname{cos}\\left(\\psi\\right)\\right) \\dot{\\theta} - \\operatorname{sin}\\left(\\beta\\right) \\operatorname{sin}\\left(\\psi\\right) \\dot{\\zeta\\left(t\\right)} - \\operatorname{cos}\\left(\\psi\\right) \\dot{\\beta})\\mathbf{j}_{A} + (\\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\zeta\\left(t\\right)\\right) \\dot{\\theta} + \\operatorname{cos}\\left(\\beta\\right) \\dot{\\zeta\\left(t\\right)} + \\dot{\\psi})\\mathbf{k}_{A}$$" - ], - "text/plain": [ - "⎛ d d d ⎞\n", - "⎜(-sin(ψ(t))⋅sin(ζ(t)) + cos(β(t))⋅cos(ψ(t))⋅cos(ζ(t)))⋅──(θ(t)) - sin(β(t))⋅cos(ψ(t))⋅──(ζ(t)) + sin(ψ(t))⋅──(β(t))⎟\n", - "⎝ dt dt dt ⎠ a_x + ⎛ d d d ⎞\n", - "⎜(sin(ψ(t))⋅cos(β(t))⋅cos(ζ(t)) + sin(ζ(t))⋅cos(ψ(t)))⋅──(θ(t)) - sin(β(t))⋅sin(ψ(t))⋅──(ζ(t)) - cos(ψ(t))⋅──(β(t))⎟\n", - "⎝ dt dt dt ⎠ a_y + ⎛ d d d ⎞\n", - "⎜sin(β(t))⋅cos(ζ(t))⋅──(θ(t)) + cos(β(t))⋅──(ζ(t)) + ──(ψ(t))⎟\n", - "⎝ dt dt dt ⎠ a_z" ] }, - "execution_count": 37, + "execution_count": 38, "metadata": {}, "output_type": "execute_result" } @@ -1063,6 +1055,13 @@ "B.ang_vel_in(A).express(A)" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "En ocasiones, la representación gráfica puede fallar, pero se puede volver a desactivar y activar llamando a la función`init_printing(pretty_print=True)` con diferentes valores (True/False) para `pretty_print`" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -1074,12 +1073,12 @@ }, { "cell_type": "code", - "execution_count": 38, + "execution_count": 39, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAASBAMAAAC3N9OQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu91EzRCrMiKZZol2\nVO8rb2IPAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAf0lEQVQIHWNgEDJgQADm/w4IDgPr/wQkHkMx\nMgeVzY+iksMfRd98MI+xgYGhBKgLwltvwFCZBOOxzl/AwHAJxuMOmoDEq2EUQOIlcCogeFwXeD6A\nefz//zMwBM2c+BUiBzSIYQYD+3c4j3cBA8MvBoZ7IAkGBnkBhsZPCzr1cwCP8yYDTbxFiQAAAABJ\nRU5ErkJggg==\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAAXCAYAAAD6FjQuAAAABHNCSVQICAgIfAhkiAAAAQxJREFUSInt088rRFEUB/APDSVs+Acm2Yhi5g+QlT9AyvbtWQ7ZW0ryH/gfLGztFcmsFUWKZmVhNxb3qWuaN29e3iDNt073/Lj3fO+551x+CQnakVTLJhgtO2EvVCL9EruR3frJiwwUif/UsyFZjAW847rDv4ozPAk93yyD7ASHWMR45J/EDXb6SZLIn8YNnGMm3bOSkevblU0IFe0JH/4ZtQLnC5Ht4wK3qd2UXVlXVPK3IDzpNpYjXxP1QZAdYxb3kW8Eb+naLkLaC+tCj+pYimQrJZnrcqbrgORVNiaM+hGuOmKP6VrDHaYwH8WrQk9beMjhAQ28Yjoj/oKDVF/z9et8ymk/REP8TXwAFeA3CvV4ZDgAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\mathbf{i}_{A1}$$" ], @@ -1087,7 +1086,7 @@ "a1_x" ] }, - "execution_count": 38, + "execution_count": 39, "metadata": {}, "output_type": "execute_result" } @@ -1099,22 +1098,35 @@ }, { "cell_type": "code", - "execution_count": 39, - "metadata": {}, + "execution_count": 40, + "metadata": { + "collapsed": true + }, "outputs": [ + { + "ename": "AttributeError", + "evalue": "'tuple' object has no attribute 'right'", + "output_type": "error", + "traceback": [ + "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[1;31mAttributeError\u001b[0m Traceback (most recent call last)", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\IPython\\core\\formatters.py\u001b[0m in \u001b[0;36m__call__\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 700\u001b[0m \u001b[0mtype_pprinters\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_printers\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 701\u001b[0m deferred_pprinters=self.deferred_printers)\n\u001b[1;32m--> 702\u001b[1;33m \u001b[0mprinter\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 703\u001b[0m \u001b[0mprinter\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mflush\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 704\u001b[0m \u001b[1;32mreturn\u001b[0m \u001b[0mstream\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mgetvalue\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\IPython\\lib\\pretty.py\u001b[0m in \u001b[0;36mpretty\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 378\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0mcls\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_pprinters\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 379\u001b[0m \u001b[1;31m# printer registered in self.type_pprinters\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 380\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_pprinters\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mcls\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcycle\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 381\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 382\u001b[0m \u001b[1;31m# deferred printer\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\interactive\\printing.py\u001b[0m in \u001b[0;36m_print_plain\u001b[1;34m(arg, p, cycle)\u001b[0m\n\u001b[0;32m 66\u001b[0m \u001b[1;34m\"\"\"caller for pretty, for use in IPython 0.11\"\"\"\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 67\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0m_can_print_latex\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 68\u001b[1;33m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtext\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mstringify_func\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\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 69\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 70\u001b[0m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtext\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mIPython\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlib\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\printing\\pretty\\pretty.py\u001b[0m in \u001b[0;36mpretty\u001b[1;34m(expr, **settings)\u001b[0m\n\u001b[0;32m 2162\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 2163\u001b[0m \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2164\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mpp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdoprint\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 2165\u001b[0m \u001b[1;32mfinally\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 2166\u001b[0m \u001b[0mpretty_use_unicode\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0muflag\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\printing\\pretty\\pretty.py\u001b[0m in \u001b[0;36mdoprint\u001b[1;34m(self, expr)\u001b[0m\n\u001b[0;32m 60\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 61\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mdoprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\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---> 62\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_print\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrender\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m**\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_settings\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 63\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 64\u001b[0m \u001b[1;31m# empty op so _print(stringPict) returns the same\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\physics\\vector\\vector.py\u001b[0m in \u001b[0;36mrender\u001b[1;34m(self, *args, **kwargs)\u001b[0m\n\u001b[0;32m 283\u001b[0m pform = vp._print(\n\u001b[0;32m 284\u001b[0m ar[i][0][j])\n\u001b[1;32m--> 285\u001b[1;33m pform = prettyForm(*pform.right(\" \",\n\u001b[0m\u001b[0;32m 286\u001b[0m ar[i][1].pretty_vecs[j]))\n\u001b[0;32m 287\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;31mAttributeError\u001b[0m: 'tuple' object has no attribute 'right'" + ] + }, { "data": { - "image/png": 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+ "image/png": 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"text/latex": [ "$$(\\operatorname{sin}^{2}\\left(\\psi\\right) \\dot{\\psi} + \\operatorname{cos}^{2}\\left(\\psi\\right) \\dot{\\psi})\\mathbf{j}_{A1}$$" - ], - "text/plain": [ - "⎛ 2 d 2 d ⎞\n", - "⎜sin (ψ(t))⋅──(ψ(t)) + cos (ψ(t))⋅──(ψ(t))⎟\n", - "⎝ dt dt ⎠ a1_y" ] }, - "execution_count": 39, + "execution_count": 40, "metadata": {}, "output_type": "execute_result" } @@ -1127,12 +1139,11 @@ }, { "cell_type": "code", - "execution_count": 40, + "execution_count": 41, "metadata": {}, "outputs": [ { "data": { - "image/png": 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"text/latex": [ "$$\\left[\\begin{matrix}0\\\\\\sin^{2}{\\left (\\psi{\\left (t \\right )} \\right )} \\frac{d}{d t} \\psi{\\left (t \\right )} + \\cos^{2}{\\left (\\psi{\\left (t \\right )} \\right )} \\frac{d}{d t} \\psi{\\left (t \\right )}\\\\0\\end{matrix}\\right]$$" ], @@ -1146,7 +1157,7 @@ "⎣ 0 ⎦" ] }, - "execution_count": 40, + "execution_count": 41, "metadata": {}, "output_type": "execute_result" } @@ -1157,12 +1168,12 @@ }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 42, "metadata": {}, "outputs": [ { "data": { - "image/png": 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"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\n", "text/latex": [ "$$\\frac{d}{d t} \\psi{\\left (t \\right )}$$" ], @@ -1172,7 +1183,7 @@ "dt " ] }, - "execution_count": 41, + "execution_count": 42, "metadata": {}, "output_type": "execute_result" } @@ -1197,7 +1208,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 43, "metadata": {}, "outputs": [], "source": [ @@ -1206,7 +1217,7 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 44, "metadata": {}, "outputs": [], "source": [ @@ -1222,7 +1233,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 45, "metadata": {}, "outputs": [], "source": [ @@ -1238,7 +1249,7 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 46, "metadata": {}, "outputs": [], "source": [ @@ -1255,12 +1266,12 @@ }, { "cell_type": "code", - "execution_count": 46, + "execution_count": 47, "metadata": {}, "outputs": [ { "data": { - "image/png": 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"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\n", "text/latex": [ "$$e_{b}\\mathbf{i}_{A1}$$" ], @@ -1268,7 +1279,7 @@ "e_b a1_x" ] }, - "execution_count": 46, + "execution_count": 47, "metadata": {}, "output_type": "execute_result" } @@ -1308,22 +1319,22 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 48, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAYBAMAAABZ8fMmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", + "image/png": 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"text/latex": [ "$$e_{b} \\dot{\\psi}\\mathbf{j}_{A1}$$" ], "text/plain": [ - " d \n", - "e_b⋅──(ψ(t))\n", - " dt a1_y" + " d\n", + "e_b⋅──(ψ(t)) a1_y\n", + " dt" ] }, - "execution_count": 47, + "execution_count": 48, "metadata": {}, "output_type": "execute_result" } @@ -1347,7 +1358,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 49, "metadata": {}, "outputs": [], "source": [ @@ -1359,12 +1370,12 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 50, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAUBAMAAACezBVvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny//EX5RzbwtR0hgsAAAAAElFTkSuQmCC\n", + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAXCAYAAADa+mvTAAAABHNCSVQICAgIfAhkiAAABA9JREFUaIHt2VuoVUUcBvCfVurhKFoGFdLloRSLolIiinpIQ7CCgnxIJIuyoJIg6CGjy0NFx+jymEJRUmgXKIiKNCiLoqI06GL1lCQqx8jTBTQ1Tw//WbXO3uuyj7qXSfuDxRzm+883s741Z/4zs+nhf4FVGER/S/0TGK54pneoU4VZSeumUY+6OfThfnyH3fgJj+CYkvhD7cOoPJqN/birgFuE1diXBDfhJTyJB0ehI9UPY2EB9xq2YWInA24YJ+Fr7BHvPoBvxLusKIjvlg8de7QWQ2IWtuJW7MXLOP0gdOAF8SIzCrgLEresbrANYxw+xx+4OFc/EZvxF05sadMtHzryaLqYTStLuL1iVtahSifDJvyOsRX8ZhzVQX9NYZkw8fYC7qnELcjVdduHWo8eTYOaU8AtTdxlFYPrRGdAeQ5elIt7INXN66C/JtCHndiqOFfeJ8a7NFfXbR/auKNbAuaKZeOTgsZjUnkF1qe4MlTpfIHnsRgfY12OW5/7+6NUXo53KvpqCtdgCp4RK1UrJqRyT66u2z5UetQvNjtfFXQOJ4hd3bBIxq/gcdw7Sh24JeksqYiZnGI+q4hpEi+K8awWG8DW59PEz0/xTfhQ6dH0RK6tEJ+bBphfHlrFOtF5OsXMroiBXdheE9MUNqs+smXPaSm+KR9GcPlEPDWVOwsa9Ytt8jqRQ67HTEzChS2xVToZzhfLVtXshV9wfE1ME+jHKeK4MqbgmSTeZwt+TG2a8mEEl8+hu1I5QTtW4GrcLHJIFap0sj7Pxrf4s0arL6d3ODEtlVtL+Hlio/Rmrq4pH0Zw+f/QwVRONRLTcB3eVf8xq3QynCleckONzlixCRmsiWsC41JZZvyNqXw2V9eED21c/oNuww7tB9zZKa6u4zqdDOemcmONzgyxnH3ZYb/dRJajWi8NiJQzH28buZ9owoc2Lv9Bh/GBWI/zt0C/pvJKHFfTeZVOhmzG/lajk+Xm9wq451I/N3QwnkOBn8UhfhbOydWfijXCo9ta2jThQxWHWFpbb0LG4sNUPySOKwO4W/uGqEonw6WJ24LHxJZ/QUFcdmd8cgG3SvsBvNtYmPrcIe6uVwo/duKikjbd9qGKQ+SK7eJMlcd43In3xXq9Jw1mP64dhU6GO/C9+KViGA+38JNFon+9pP1GMbOPLXuRLmGxuIjfLT7ESv9umIrQTR/qPPoH9yTx82ri5qS4NQepU4TsmvGSAm6KuH1ZfgC6hwPd8qGKG4EJ4hD9Rk3czCRYZmynOq3oE8eDV0v4q8SMLtqg/BfRDR9KuaJb+n1iRzte/FRUdG+5RJxNh8XZdOgAdYpwhjgeLC/R/QEPiZ+wjgR0w4c6j0aNDXgLZx0KsR566KGHHno4AvE3thd9mKFEsMEAAAAASUVORK5CYII=\n", "text/latex": [ "$$\\left ( \\xi{\\left (t \\right )}, \\quad \\theta{\\left (t \\right )}\\right )$$" ], @@ -1372,7 +1383,7 @@ "(ξ(t), θ(t))" ] }, - "execution_count": 49, + "execution_count": 50, "metadata": {}, "output_type": "execute_result" } @@ -1385,7 +1396,7 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 51, "metadata": {}, "outputs": [], "source": [ @@ -1396,12 +1407,11 @@ }, { "cell_type": "code", - "execution_count": 51, + "execution_count": 52, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAABLCAMAAABqUbriAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRIki781m3btsR7x/vAAAAAlwSFlzAAAOxAAADsQBlSsOGwAABnpJREFUeAHtXIuS\nrCgMxefe9e36//+6EEhABBXMOHNr2qqRdIQcDgHUpHtEscFRil9yLJqvEMVW1fJofglvMSu25aaI\nF+mch1m3aWcjCNRI/TwYgySkA7gt0LQFC6Elgc2ZxIdVd6xb26YCETXwoUTmJLg8YnIx1sFLaNqC\nCVTt0M7B2qqfe3JxLvFJMyvGVohN/glhNO2irrQT9MgRzOfTotj64PUjWBiNUMNW5FrWvVOXM4nP\n2stiUfN8U7sDaho5heTRo+9IAHXWCU1bsBjaGdg8Qr/QI5nEwa2SLRjbFHujEZW23MIVqSchizM0\nOoLF0M7AVrhrNXp65np8WDSNRTm+UB5HjVjMbJ06w5QE8/mkGApagk4tNG3B4mgnYBsQ72B6Suth\nj8t9oK/UBG77uu4VhaJu5hqniaj1TO82dalWZ63pynVbS5jllVkMggRZNXIY48OqOldMY1XI2w2u\nlQBYHC0O1m7QoW4zdoPEh1GSLsZBFDDL1lq0sIkbPwux6ltYuZXyGNWqNhphlrgcURwlEiKs5eiS\n8Qm8skhAMWx4YziCxdHiYIPeOGn/DBKfVKcbeXfX81be5zu1ewtywqJmg1zXQG5TZ6PBJS6b4yiR\nAE1CJ2tcr8PV7hyq+hEMVUe0ONigPX5K3MwKs3oltBysZVtrdIEQo17AMIiwxFFDS1yufMORBPM5\nUJBxQ1yPJw7zESyOFge7M9VpHTSwhiXxVS72adNjpjqu+9LCRlGBf0zvWtNCDZphSEKAsVGRcUMc\npj4uxgAYEj+inYDpza0529yOHq8GNdNnYqVnn8bVHTTzsVN04Xmmw6lOQpS4NR4kfgTDqX5EOwFb\nYTjn09sZrHG5TMVodsKugXlX4mO52W+Um3tN0Gh6OUsbuCfRNkNClLg1HiR+BMPN7Yh2AqafOcxj\nRuR2NoyKYS939VGRmHrRwOa24irv9b2qqsUg9351oKYUrV6bFT59kgD1QidrXHtFn2mqo2kLFkc7\nA1MzpwVCqhPBXV20ZVVL3vKmIm/opfRr18/1DDd06LmZUW1VVzCvZQXt+GHFl7UJn0VIgJahExov\nym0rWzgXzbqNcG8j0w5YHO0MTD2TlNirCPFQ93Y6fEC1Sk+Dj1v2IctWTZU806q5p0I0LG8ghD1+\n2RDfG2xFT0OvCyTYqqmSZ1o191QIguUNhEzigpY7Yew09IJIAtXLEHamdfudCkGwvAORS3wXCACg\nnYb6RcKdzsTq7EzrSjsVgmAZs+Pqc4m7oR9jD+ND8iMFgUhwMdNlxzQ2dlQIgiVWOS2ziZ9a/Qsu\nfoj/BU5i7eLH46zD+RcYe+xx3F0ZY/2BYeNHuUv8a2P9Aao7Fd61H2UU8hIKFLLZdYjSCE5iISfW\n7xk9fGTJKEzq7Ywe8u96/NAXrcBn5mex/ohxUrOg8CQUsE9mAJ3EAr04eZmFs1g/WouVLCg6yJGc\nUPjSWH+MsNHjy+azjMKthILtyRuxfnGRvzimLzIyCneirJb2S7H+i/wFS0bhVkLBMn8l1n+VvzAR\nXCd9gWHWhIzCaUKhnRY6TPTqhVj/MZrt5S9YMgqhqf7Pn3+tjz3phVj/Zf7imL7IySh4CYX//qik\nB8UePdrijVj/0eNe/oIno3AnoWDpvxLrv8pf8GQU7iQUHOKUSPjCWP9V/oIpo3AnoUDM34n1X+Qv\njumLrIwCS0IBR4ae+VHBEeu3toz0FShMLylOX/GNwqgwxo+lU/O26JlU7TwVWsfy2vRD4oHEwl6F\nMX4sr3sUqhEImO9UaB3LkA1P95Q4hggcszsV9g9Lp16CuDOp2+1UaB3LG6afEg8kFlwVxvixvNGj\nYBUMPTkXHRVax9KpFRUfE49a/uEXPsR/uIPYu/fxOPuQ/nCDH48nO2gfn09untqAGy7f4/v4fCqP\n5PoccPp7ZRo6m7gXn08mktjgOZx8BSzxa40SPJu4F59P5JFcnQWu5iDuxeeTmSQ2YIHjIO4HLRN5\npFbngeMg7sfnU5kk1ueB4yEO3+ONZY8TeV1W99MBlw2CFTiI88y9YPdCSh44DuLCi8+HesupY4Fj\nIe7F5zlJhmyxwLEQ9+Lzoc5y6ljgWIhDwtJ+4Z+TZNDWPh0QrHKp5CG+j89fgj6t8Byukj8NnOhn\nNdmPrE+JfHf7D/Hv9sDb+B+Pvz3i3433yz3+S/8BTqv+IUxd659Gf/cUfAEf/gFOXYv/Aa09Y1Fl\nQuM/AAAAAElFTkSuQmCC\n", "text/latex": [ "$$\\left[\\begin{matrix}\\cos{\\left (\\theta{\\left (t \\right )} \\right )} & \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\- \\sin{\\left (\\theta{\\left (t \\right )} \\right )} & \\cos{\\left (\\theta{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$" ], @@ -1413,7 +1423,7 @@ "⎣ 0 0 1⎦" ] }, - "execution_count": 51, + "execution_count": 52, "metadata": {}, "output_type": "execute_result" } @@ -1424,7 +1434,7 @@ }, { "cell_type": "code", - "execution_count": 52, + "execution_count": 53, "metadata": {}, "outputs": [], "source": [ @@ -1436,20 +1446,20 @@ }, { "cell_type": "code", - "execution_count": 53, + "execution_count": 54, "metadata": {}, "outputs": [ { "data": { - "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAATBAMAAAAdcyJ3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMmaruyJ271SZ\niUSa32RUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA50lEQVQYGWNgQAMciwVQRVjzv6WXBzDw/A9A\nFWfgL2Dg+MbA+t8BTdx/AgNDvgPDPjRhhvlAhfYN6KIMDFUMDNzfGd7/v4Am9Y2BYVcCA4c9mjjL\nt5n1R4BK6y8w3KkRQOjh/MjAqggW521gU2AQMgYBFQYG9gUMDH/B4swNLD8Q6vmBZgOtAJrD/IDj\nH0LcP4CB+w9YHOiujwhxoPN5fzIIgOxlYJ6AENdnYGD9ybIBLH4HIXzofyoDw/GZYHeyHkCIQ1ki\nIPVvGZ6gS/x3WL+BRXt3IYb4k88MXP//f0AXf2R8ECwEAJWaRfWXrUh4AAAAAElFTkSuQmCC\n", + "image/png": "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\n", "text/latex": [ "$$- R\\mathbf{j}_{2}$$" ], "text/plain": [ - "- R 2_y" + "-R 2_y" ] }, - "execution_count": 53, + "execution_count": 54, "metadata": {}, "output_type": "execute_result" } @@ -1464,24 +1474,35 @@ }, { "cell_type": "code", - "execution_count": 54, - "metadata": {}, + "execution_count": 55, + "metadata": { + "collapsed": true + }, "outputs": [ + { + "ename": "AttributeError", + "evalue": "'tuple' object has no attribute 'right'", + "output_type": "error", + "traceback": [ + "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[1;31mAttributeError\u001b[0m Traceback (most recent call last)", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\IPython\\core\\formatters.py\u001b[0m in \u001b[0;36m__call__\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 700\u001b[0m \u001b[0mtype_pprinters\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_printers\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 701\u001b[0m deferred_pprinters=self.deferred_printers)\n\u001b[1;32m--> 702\u001b[1;33m \u001b[0mprinter\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 703\u001b[0m \u001b[0mprinter\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mflush\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 704\u001b[0m \u001b[1;32mreturn\u001b[0m \u001b[0mstream\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mgetvalue\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\IPython\\lib\\pretty.py\u001b[0m in \u001b[0;36mpretty\u001b[1;34m(self, obj)\u001b[0m\n\u001b[0;32m 378\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0mcls\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_pprinters\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 379\u001b[0m \u001b[1;31m# printer registered in self.type_pprinters\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 380\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtype_pprinters\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mcls\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcycle\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 381\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 382\u001b[0m \u001b[1;31m# deferred printer\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\interactive\\printing.py\u001b[0m in \u001b[0;36m_print_plain\u001b[1;34m(arg, p, cycle)\u001b[0m\n\u001b[0;32m 66\u001b[0m \u001b[1;34m\"\"\"caller for pretty, for use in IPython 0.11\"\"\"\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 67\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0m_can_print_latex\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 68\u001b[1;33m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtext\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mstringify_func\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\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 69\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 70\u001b[0m \u001b[0mp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mtext\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mIPython\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlib\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mpretty\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0marg\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\printing\\pretty\\pretty.py\u001b[0m in \u001b[0;36mpretty\u001b[1;34m(expr, **settings)\u001b[0m\n\u001b[0;32m 2162\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 2163\u001b[0m \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2164\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mpp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mdoprint\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 2165\u001b[0m \u001b[1;32mfinally\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 2166\u001b[0m \u001b[0mpretty_use_unicode\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0muflag\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\printing\\pretty\\pretty.py\u001b[0m in \u001b[0;36mdoprint\u001b[1;34m(self, expr)\u001b[0m\n\u001b[0;32m 60\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 61\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mdoprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\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---> 62\u001b[1;33m \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_print\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrender\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m**\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_settings\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m 63\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m 64\u001b[0m \u001b[1;31m# empty op so _print(stringPict) returns the same\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;32m~\\Documents\\Python\\condaenv\\lib\\site-packages\\sympy\\physics\\vector\\vector.py\u001b[0m in \u001b[0;36mrender\u001b[1;34m(self, *args, **kwargs)\u001b[0m\n\u001b[0;32m 283\u001b[0m pform = vp._print(\n\u001b[0;32m 284\u001b[0m ar[i][0][j])\n\u001b[1;32m--> 285\u001b[1;33m pform = prettyForm(*pform.right(\" \",\n\u001b[0m\u001b[0;32m 286\u001b[0m ar[i][1].pretty_vecs[j]))\n\u001b[0;32m 287\u001b[0m \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n", + "\u001b[1;31mAttributeError\u001b[0m: 'tuple' object has no attribute 'right'" + ] + }, { "data": { - "image/png": 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"text/latex": [ "$$(R \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta} + \\dot{\\xi})\\mathbf{i}_{0} + R \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}\\mathbf{j}_{0}$$" - ], - "text/plain": [ - "⎛ d d ⎞\n", - "⎜R⋅cos(θ(t))⋅──(θ(t)) + ──(ξ(t))⎟\n", - "⎝ dt dt ⎠ 0_x + d \n", - "R⋅sin(θ(t))⋅──(θ(t))\n", - " dt 0_y" ] }, - "execution_count": 54, + "execution_count": 55, "metadata": {}, "output_type": "execute_result" } @@ -1574,7 +1595,7 @@ }, { "cell_type": "code", - "execution_count": 55, + "execution_count": 56, "metadata": {}, "outputs": [ { @@ -1599,7 +1620,7 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 57, "metadata": {}, "outputs": [ { @@ -1771,7 +1792,7 @@ "" ] }, - "execution_count": 56, + "execution_count": 57, "metadata": {}, "output_type": "execute_result" } @@ -1800,7 +1821,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.4" + "version": "3.6.4" } }, "nbformat": 4, diff --git a/notebooks_vacios/040-SymPy-Intro.ipynb b/notebooks_vacios/040-SymPy-Intro.ipynb index 4bf15c6..3b15c79 100644 --- a/notebooks_vacios/040-SymPy-Intro.ipynb +++ b/notebooks_vacios/040-SymPy-Intro.ipynb @@ -109,7 +109,24 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Lo primero, como siempre, es importar aquello que vayamos a necesitar:" + "Lo primero, como siempre, es importar aquello que vayamos a necesitar. La manera usual de hacerlo con SymPy es importar la función `init_session`:\n", + "```\n", + "from sympy import init_session\n", + "init_session(use_latex=True)```\n", + "\n", + " Esta función ya se encarga de importar todas las funciones básicas y preparar las salidas gráficas. Sin embargo, en este momento, esta función se encuentra en mantenimiento para su uso dentro de los notebooks por lo que activaremos la salida gráfica e importaremos las funciones de la manera usual. Puedes consultar el estado de la corrección en: https://github.com/sympy/sympy/pull/13300 y https://github.com/sympy/sympy/issues/13319 ." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "El comando `init_session` llevaría a cabo algunas acciones por nostros:\n", + "\n", + "* Gracias a `use_latex=True` obtenemos la salida en $\\LaTeX$.\n", + "* __Crea una serie de variables__ para que podamos ponernos a trabajar en el momento.\n", + "\n", + "Estas capacidades volverán a estar disponibles cuando el problema se corrija." ] }, { @@ -117,9 +134,7 @@ "execution_count": null, "metadata": {}, "outputs": [], - "source": [ - "# Importación" - ] + "source": [] }, { "cell_type": "code", @@ -129,13 +144,15 @@ "source": [] }, { - "cell_type": "markdown", + "cell_type": "code", + "execution_count": 5, "metadata": {}, + "outputs": [], "source": [ - "Lo primero que vemos es que el comando `init_session` ha llevado a cabo algunas acciones por nostros:\n", - "\n", - "* Gracias a `use_latex=True` obtenemos la salida en $\\LaTeX$.\n", - "* __Ha creado una serie de variables__ para que podamos ponernos a trabajar en el momento." + "# aeropython: preserve\n", + "from sympy import (symbols, pi, I, E, cos, sin, exp, tan, simplify, expand, factor, collect,\n", + " apart, cancel, expand_trig, diff, Derivative, Function, integrate, limit,\n", + " series, Eq, solve, dsolve, Matrix, N)" ] }, { @@ -1184,7 +1201,7 @@ }, { "cell_type": "code", - "execution_count": 77, + "execution_count": 78, "metadata": {}, "outputs": [ { @@ -1356,7 +1373,7 @@ "" ] }, - "execution_count": 77, + "execution_count": 78, "metadata": {}, "output_type": "execute_result" } @@ -1385,7 +1402,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.4" + "version": "3.6.4" } }, "nbformat": 4, diff --git a/notebooks_vacios/041-SymPy-Mecanica.ipynb b/notebooks_vacios/041-SymPy-Mecanica.ipynb index efb1252..0f4638f 100644 --- a/notebooks_vacios/041-SymPy-Mecanica.ipynb +++ b/notebooks_vacios/041-SymPy-Mecanica.ipynb @@ -37,6 +37,25 @@ "El objeto primordial que vamos a manejar van a ser los sistemas de referencia. Podremos definir relaciones geométricas entre ellos y de esta forma las transformaciones de vectores entre un sistema y otro serán triviales." ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "La manera usual de empezar a trabajar con SymPy es importar la función `init_session`:\n", + "```\n", + "from sympy import init_session\n", + "init_session(use_latex=True)```\n", + "\n", + " Esta función ya se encarga de importar todas las funciones básicas y preparar las salidas gráficas. Sin embargo, en este momento, esta función se encuentra en mantenimiento para su uso dentro de los notebooks por lo que activaremos la salida gráfica e importaremos las funciones de la manera usual. Puedes consultar el estado de la corrección en: https://github.com/sympy/sympy/pull/13300 y https://github.com/sympy/sympy/issues/13319 ." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + }, { "cell_type": "code", "execution_count": null, @@ -107,7 +126,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 7, "metadata": {}, "outputs": [], "source": [ @@ -455,10 +474,19 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "En ocasiones, la representación gráfica puede fallar, pero se puede volver a desactivar y activar llamando a la función`init_printing(pretty_print=True)` con diferentes valores (True/False) para `pretty_print`" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -478,7 +506,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "#v1.diff(dynamicsymbols._t, A2)" @@ -674,7 +704,9 @@ { "cell_type": "code", "execution_count": null, - "metadata": {}, + "metadata": { + "collapsed": true + }, "outputs": [], "source": [ "# Hallamos la velocidad de P en A1, expresada en A0\n", @@ -770,7 +802,7 @@ }, { "cell_type": "code", - "execution_count": 56, + "execution_count": 57, "metadata": {}, "outputs": [ { @@ -942,7 +974,7 @@ "" ] }, - "execution_count": 56, + "execution_count": 57, "metadata": {}, "output_type": "execute_result" } @@ -971,7 +1003,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.4" + "version": "3.6.4" } }, "nbformat": 4,