sphere.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from GRpy.all import *\n",
    "init_printing(use_unicode=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sphere:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Metric:\n",
    "$$\n",
    "g_{ab}(\\theta, \\phi)=\n",
    "\\begin{bmatrix}\n",
    "a^2 & 0\\\\\n",
    " 0 & a^2 \\sin^2(\\theta)\n",
    "\\end{bmatrix},\n",
    "$$\n",
    "with $a$ the radius of the sphere."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First we define the symbols for the coordinates $x^a=(\\theta,\\phi)$ and the constants:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a, theta, phi = symbols('a, theta, varphi', real=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we define the variable `gab` as a Metric depending on the coordinates, and asign the components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "gab = Metric((theta, phi))\n",
    "gab[-0,-0] = a**2\n",
    "gab[-0,-1] = 0\n",
    "gab[-1,-0] = 0\n",
    "gab[-1,-1] = (a**2)*(sin(theta))**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The object `gab` is of type `GRpy.Tensor.Metric`, and has several associated methods:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GRpy.Tensor.Metric"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(gab)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlMAAAAaCAYAAACTidllAAAABHNCSVQICAgIfAhkiAAAChlJREFU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\nFEVRFEVRFEVRFEVRmsH/AeVnKpLNuoLBAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left \\{ \\left ( -1, \\quad -1\\right ) : a^{2} \\sin^{2}{\\left (\\theta \\right )}, \\quad \\left ( -1, \\quad 0\\right ) : 0, \\quad \\left ( 0, \\quad -1\\right ) : 0, \\quad \\left ( 0, \\quad 0\\right ) : a^{2}\\right \\}$$"
      ],
      "text/plain": [
       "&#9127;           2    2                                      2&#9131;\n",
       "&#9128;(-1, -1): a &sdot;sin (&theta;), (-1, 0): 0, (0, -1): 0, (0, 0): a &#9132;\n",
       "&#9129;                                                        &#9133;"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gab.components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAAaCAYAAABRj+AQAAAABHNCSVQICAgIfAhkiAAACQNJREFU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\neDwej8fj8Xg8Ho/H4/F4PB6Px9N//A/pFToHRlbwhQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( -1, \\quad -1\\right ), \\quad a^{2} \\sin^{2}{\\left (\\theta \\right )}\\right ), \\quad \\left ( \\left ( 0, \\quad 0\\right ), \\quad a^{2}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;           2    2   &#9118;  &#9115;         2&#9118;&#9124;\n",
       "&#9123;&#9117;(-1, -1), a &sdot;sin (&theta;)&#9120;, &#9117;(0, 0), a &#9120;&#9126;"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gab.getNonZero()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAARCAYAAAAougcOAAAABHNCSVQICAgIfAhkiAAAAYdJREFU\nOI3t1D1IllEUB/BfpZlGUoN9QF+L2NQiRYtUQxkhES0VUVNDDVE0CDnYIiJIIEJNEdEQOSYNRUPZ\nBxTUUEG01BIVlIQQplhiwz0P3h5e0uHFqf9yzzn3nPO/555zLwuARRVsS9GFZnzDO+zBbdyoBmk9\nHmIos13Ab5yqBgEM4isaM9s+zKClGgQb8AsDJXsPvswj/kT4tf3L6Zx04vaS/TFuzoNkOUZRW95Y\nnMlbYn2e2ZZhm9SnubALj6Tb+As1mTyGH7EW2Ik6PMj8u/FRGpK9OIv3IU/jMHbjCl6XCVulKVod\n+iZ8wKfM52okhTWYkKqFt9gRcgeGK1XyEqdxLQJ+4rv0TmArjuBMpj/BJNZHrmextxZN5SoqoQFT\nOBn6edzN9vvQiVU4GIcrMISLhVI0vgkHSiT7sQT3Qh8zO8orcAgjOIZxs71sjiovFYmKb+VWBK2U\nrmkdnko96A2felzG/ZA3So1+EQfpxxtsl97W5zLJURzHK2nON+M67viPhcYfdXRM8EZh4nAAAAAA\nSUVORK5CYII=\n",
      "text/latex": [
       "$$g_{ab}$$"
      ],
      "text/plain": [
       "g_{ab}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gab.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "======================================================================\n",
      "The non-zero components of g_{ab} are:\n",
      "(-1, -1) : a**2*sin(theta)**2\n",
      "(0, 0) : a**2\n",
      "======================================================================\n"
     ]
    }
   ],
   "source": [
    "gab.__str__()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inverse is computed using `.invert()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "ginv = gab.invert()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAAhCAYAAACsuqLcAAAABHNCSVQICAgIfAhkiAAACKtJREFU\neJztnXmsHVMcxz9PH6qlWqqljaKP1lIlKLE1GkJQS4SgtVxBq9ZEKkSJR+3EHk2qTYoIIaVF7eo2\nKap21WqqTbTW0tLiRau2P34z6bxx5txzzsydmXff+SQv9945c8/9zrvf/O5ZfwMej8fj+R+jgUXA\n3ODvAM25w4E78hDlSUU/YMuiRRjiPeVxwdTjD7Ixtn2sO7ES/NViKDATaDY415M9WwHTgQEG5/YD\nngzeU2a8pzo+Nr7MEhePV3WFFWoHwm7Ae+R/sR7hAuBG4F9gZ8P37A08BzTVSVNavKc6Pi6+zBJb\nj1d1hRVqB8KbgGsMP8xTP2wNNwEYXx8pqfGeahyKCoRg5/GqrrCCPhD2Ab5G3wQdAJwVOzYYeBa4\nF7gHeAzoq5VZf2pp6gLcD3TPX5oRtobbHFgM9K+LGndcPRXSDfgsa1EpSNJTdj9lRZGB0MbjVV1h\nBX0gnAA8oCnfGngc2DR27BvaG/la4HNEeBGYahqEBMgy4mK4K4DJ2UtJhYunQoYB7yP/izJQS0+Z\n/ZQVRQZCMPd4VVdYQR8IlwEHa8ofBPaLHbsF+JH2g+DbABuAcToxdcRGUytwdj6yrHAx3DbAGqBX\n5mrccfHUHsAsYBowj+IDoY2eVsrpp6woOhCaeryqK6yQHAiHAL8AmySUDwTeUBxfArygOL4AmK0T\nU0dsNPVGujpd6i3KElfDvQSMyVaKM66eijKN4gNhlGno9ZTVT1lRdCAEM49XwydJ5kviaGT9zT8J\n5ZcgXZgoWwG7ASsU538H7G+pIQtsNa0CvgVG1FlXXswGRhYtIsDFUx2dRvNTGbHyuG0gHIZ+UPo4\nxNRRdgoe1yrObwN6AJtZ6kiLi6a3gRPrKcqC0cCk4PntwKWW738XWbhcBlw81QiUyU9ZkdaXWWLl\ncdtAuDcyI6OiPzL7tyx2vEfwuEHxnrbgsaeljrS4aPqEYlqvKp5AxjGbgDOAhyzfvxCZgBiYsS4X\nXDzVCJTJT1mR1pdZYuVx20A4AFidULYzMvkQ5+/gUTVmEs4C5r2TwEXTaqQ73QisQQJ+Ga7HxVON\nQCP5qYxYedwmEG6JjK39mlDeF3VX8ydNneF6qt8sdGSBi6afkV+YRmEtsGPBGlw91Qg0mp/KiLHH\nbQJhGCCSTNuMesB7JdLyUk1ld0cid96B0EVTM7C+zrryZA2yzKBIXD3VCDSan8qIscdtuqThIuO2\nhPKfUAeWNiTLgyoy74qMleSNi6ZeFNNNS7ssJGnf5Vpgi5R1p8XVU41AR/VTWVH53NjjNoEw/PXq\nllD+LbBtQtks4HxEbPhFtCCB6PbYuYOB5cA6C20u2GgC2A7ZBhYlD621NpBvgiwxCb/wOw3rTWpt\n5fX/h3SesiHPazJF5Seov1aThASunjKhnnXHMe5R2HSNfw8ek8Y1vkQmIfopyiYh3aDodrbLkO1s\nUyLHRiAziE9Y6HLFVFPIMCQ7SkieWnWMBGYghjoI85nInsAfsWN5X1MaT0XpGjyqAmoR35NOT0jc\nT9DxPVV03XFUHldiEwjbkHGzHgnl/wKvAYcryr4HjkCm1O8FHkFaXscCf0bOW4ksNs1jWYGpppDD\nkFx5IXlq1TEQODN4vgzzCZCt+X/XLO9rSuOpPsCryA/X6cGx5cBbyHq2kLyuyVRPSNxP0PE9VXTd\ncVQeN6ZC8ha7BSRnAAE4BHjG9YMjtGZQR5a0AB8mlLXmqEPF5mzM2vIytVtPIetI3tnQmlKTDZ3R\nUzo/QfFaXT1VdN1xdB6HFFvsFiBjGEm8g6zDa7GsN05RGWmSuBLJTKOiaK3rkVbVocAcZItgLfoj\nuhcllOd5TZ3RUzo/QfFaXTyVVd0noZ+7GIQswq9FLY+3wzYQzkfSqesYD0zEPRPyCPS/lnkzHNmB\n8qqirCxaeyBaVJM8KvZCJiJWKsryvqbO5imdn6A8Wm089RFmt/eoVfexwFLgr8ix7sD1Qf23IslS\nhlN7N5rO40ZUSL6oPZGFoLUC6GHARQ6f3Qzc5vC+etEFuA51lpAyaR2DtJo2BY4yOH8i8KjieBHX\n1Jk8pfMTlEurjadOwC7Zb1Ld8WtvQlZ3hGOmTyItwhZkYlNHksejVHWFFfTRfQky2+MpB6cjC5JX\nIemshhi8Zx5waj1FWeI9VS5cPJW27s2QScso5wOvRF7PRJJwgD6RL5h5vKorrKAPhFcj6cY9+dGM\n3NPjQuBy4EXcx8xakO5C3hl/dHhP5Yurn/oi45unIastpiEtuvnIdwjSOvwcmfCpAOciN1SqdXuC\nHYCbI6+bkG7yaZFjK5CxRYCpmrpMPV7VFVbQB8LewFd0nPvkNgJTkPTjIGb8g41r1Wy5C7ghC1EZ\n4j2VL65+Go+00kLCxKcTaT/TfTPSpQ15HjilRt1dgfsirw9AxlJbkZt63RnoDD2iS8Vv6vFq+MR2\nsgSkSTsVuNjhvR57hiK/vpMjr+fitvOgD3AycHc20jLDeyo/0vhpFnK7zvnIzc5eCo7/HTvvL2QC\nJeRnkteKhqyj/STJgcHntCKTKj8ATyGL8HcHPkiox8njLoEQJOKeSPHZSzoDRyFGDVfIHwm8jtse\n3PuAq0je21sk3lP5kMZPG4BdEA9tQFpUSd3PeHA0YT6y1x8kcIbBtAlpiYYtxmNI3n3j5HHXQLgO\nGV+4n/xzCXY21iC7YEAWop6CrL9S7VbQMQpZszUjO2mZ4j2VD2n8NBpZljIHGRP8FPchGhVPIzfp\nakZ2nYRbMM8DpgefNwjZraMKdM4eV03j74us2xmF3Ln+U9QLKlchg5fjgDdtP9hjzGJkpqwbsA/y\nPx+IZOBdaljHMOB4ZKC7zHhP1Z80fjoImeXdMXi+BElzNRZpyX0RlI9FAtY3SBf3HOT2GEtRJ5qI\nshBJlLAEmSjZHhkXvCMoX416baCpxx9G4tsFSOKLIrNoezwej8fj8Xg8Ho/HUxb+Aw+ELhhjcZCx\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( 0, \\quad 0\\right ), \\quad \\frac{1}{a^{2}}\\right ), \\quad \\left ( \\left ( 1, \\quad 1\\right ), \\quad \\frac{1}{a^{2} \\sin^{2}{\\left (\\theta \\right )}}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;        1 &#9118;  &#9115;            1     &#9118;&#9124;\n",
       "&#9122;&#9116;(0, 0), &#9472;&#9472;&#9119;, &#9116;(1, 1), &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9119;&#9125;\n",
       "&#9122;&#9116;         2&#9119;  &#9116;         2    2   &#9119;&#9125;\n",
       "&#9123;&#9117;        a &#9120;  &#9117;        a &sdot;sin (&theta;)&#9120;&#9126;"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ginv.getNonZero()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "======================================================================\n",
      "The non-zero components of g_inv are:\n",
      "(0, 0) : a**(-2)\n",
      "(1, 1) : 1/(a**2*sin(theta)**2)\n",
      "======================================================================\n"
     ]
    }
   ],
   "source": [
    "ginv.__str__()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The method `.invert()` computes the inverse:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Christoffel symbols of the second kind $\\{^a_{bc}\\}$ are computed using the function `Christoffel`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "chris = Christoffel(gab)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAAYCAYAAACSuF9OAAAABHNCSVQICAgIfAhkiAAAAaVJREFU\nSInt1j9oVEEQx/GPl4j/sLCJoGIjdsHGKDYBRZMqEFEERbAQwTIWYplKbMQiTYRgYaWNaVIF0uSP\noLYKFkKCIEbBIFqcF/FPUuwc9wwHd7l3L1jcr9mZfTtvv+8xuzP8Z9oW43ncxnHsxFe8wl9sx370\nogsHsRxx3RjFB+zCIEaw2C7Am1jD3TrPjmAJpzJzjwJAQFfig1pWaYP/M8bfddYuSqCHwz+Gy5jI\n+M+x2k6gRlrIAJ0LgEr4ZzGDfVsJtISVsL/hU9h7cQFzuLqVQH/wOOynUr5dwSU8kZI6V0J354it\n4Hqezetps3+ocHWAGqlooGvSSexvNqBooEmp9LxsNmAjUPXa390moNOYx69mA6rFdVgqrn0BtYbX\n+IyLKLcINIYDeIYzGI/3lnBLKt7fY8+HLe6xKb1VK8ZDmAp7AnfCHpZKUuE6hHcZ/wZe4AS+YEfM\n70FPdVGem7qR+qTiW9UApqW8mlXrLMoyKVHkKStLBRiOSu3JA3z0b052SdcDakldhEq4jzc4KfVS\ny7HnPbzHD+lET6p1ER11lEvricVNfBFNXekAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\Gamma^{a}_{\\ \\ bc}$$"
      ],
      "text/plain": [
       "\\Gamma_{\\ \\ bc}__{a}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chris.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvUAAAAgCAYAAACSLoc4AAAABHNCSVQICAgIfAhkiAAADzdJREFU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\nFEVRFEVRFEVRFEVRFOB/gcXE+D8uoasAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( 0, \\quad -1, \\quad -1\\right ), \\quad - \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )}\\right ), \\quad \\left ( \\left ( 1, \\quad -1, \\quad 0\\right ), \\quad \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}}\\right ), \\quad \\left ( \\left ( 1, \\quad 0, \\quad -1\\right ), \\quad \\frac{\\cos{\\left (\\theta \\right )}}{\\sin{\\left (\\theta \\right )}}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;                               &#9115;            cos(&theta;)&#9118;  &#9115;            cos(&theta;)&#9118;&#9124;\n",
       "&#9122;((0, -1, -1), -sin(&theta;)&sdot;cos(&theta;)), &#9116;(1, -1, 0), &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9119;, &#9116;(1, 0, -1), &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9119;&#9125;\n",
       "&#9123;                               &#9117;            sin(&theta;)&#9120;  &#9117;            sin(&theta;)&#9120;&#9126;"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chris.getNonZero()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we want access to a particular component, we can write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGsAAAASCAYAAABckiAFAAAABHNCSVQICAgIfAhkiAAAA+JJREFU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\nB3TeKz353lvdPWnvRz/60Y//I/4B6OkWbWpHmJoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "-sin(&theta;)&sdot;cos(&theta;)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chris.components[(0,-1,-1)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, the Riemann tensor can be computed from the Christoffel symbols:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "riem = Riemann(chris)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACwAAAAWCAYAAAC7ZX7KAAAABHNCSVQICAgIfAhkiAAAAlpJREFU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\n",
      "text/latex": [
       "$$R_{abcd}$$"
      ],
      "text/plain": [
       "R_{abcd}"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "riem.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABSYAAAAaCAYAAACq0oI6AAAABHNCSVQICAgIfAhkiAAADVdJREFU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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( -1, \\quad 0, \\quad -1, \\quad 0\\right ), \\quad a^{2} \\sin^{2}{\\left (\\theta \\right )}\\right ), \\quad \\left ( \\left ( -1, \\quad 0, \\quad 0, \\quad -1\\right ), \\quad - a^{2} \\sin^{2}{\\left (\\theta \\right )}\\right ), \\quad \\left ( \\left ( 0, \\quad -1, \\quad -1, \\quad 0\\right ), \\quad - a^{2} \\sin^{2}{\\left (\\theta \\right )}\\right ), \\quad \\left ( \\left ( 0, \\quad -1, \\quad 0, \\quad -1\\right ), \\quad a^{2} \\sin^{2}{\\left (\\theta \\right )}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;                 2    2   &#9118;  &#9115;                  2    2   &#9118;  &#9115;               \n",
       "&#9123;&#9117;(-1, 0, -1, 0), a &sdot;sin (&theta;)&#9120;, &#9117;(-1, 0, 0, -1), -a &sdot;sin (&theta;)&#9120;, &#9117;(0, -1, -1, 0),\n",
       "\n",
       "   2    2   &#9118;  &#9115;                 2    2   &#9118;&#9124;\n",
       " -a &sdot;sin (&theta;)&#9120;, &#9117;(0, -1, 0, -1), a &sdot;sin (&theta;)&#9120;&#9126;"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "riem.getNonZero()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, `Riemann()` computes the covariant components of the curvature. The completely contravariant components $R^{abcd}$ can then be computed using the method `.Riemann_cont`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAWCAYAAABQUsXJAAAABHNCSVQICAgIfAhkiAAAAkhJREFU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\nkWp7jmIJJZTwHeYzfD6rUhAeAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$R^{abcd}$$"
      ],
      "text/plain": [
       "R__{abcd}"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "riem.Riemann_cont.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABC0AAAAhCAYAAAD9AVFjAAAABHNCSVQICAgIfAhkiAAADdtJREFU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\nCCGEEEII+/wfGqL/Yi4sTYUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( 0, \\quad 1, \\quad 0, \\quad 1\\right ), \\quad \\frac{1}{a^{6} \\sin^{2}{\\left (\\theta \\right )}}\\right ), \\quad \\left ( \\left ( 0, \\quad 1, \\quad 1, \\quad 0\\right ), \\quad - \\frac{1}{a^{6} \\sin^{2}{\\left (\\theta \\right )}}\\right ), \\quad \\left ( \\left ( 1, \\quad 0, \\quad 0, \\quad 1\\right ), \\quad - \\frac{1}{a^{6} \\sin^{2}{\\left (\\theta \\right )}}\\right ), \\quad \\left ( \\left ( 1, \\quad 0, \\quad 1, \\quad 0\\right ), \\quad \\frac{1}{a^{6} \\sin^{2}{\\left (\\theta \\right )}}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;                  1     &#9118;  &#9115;                 -1     &#9118;  &#9115;                 -1 \n",
       "&#9122;&#9116;(0, 1, 0, 1), &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9119;, &#9116;(0, 1, 1, 0), &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9119;, &#9116;(1, 0, 0, 1), &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "&#9122;&#9116;               6    2   &#9119;  &#9116;               6    2   &#9119;  &#9116;               6    \n",
       "&#9123;&#9117;              a &sdot;sin (&theta;)&#9120;  &#9117;              a &sdot;sin (&theta;)&#9120;  &#9117;              a &sdot;sin\n",
       "\n",
       "    &#9118;  &#9115;                  1     &#9118;&#9124;\n",
       "&#9472;&#9472;&#9472;&#9472;&#9119;, &#9116;(1, 0, 1, 0), &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9119;&#9125;\n",
       "2   &#9119;  &#9116;               6    2   &#9119;&#9125;\n",
       " (&theta;)&#9120;  &#9117;              a &sdot;sin (&theta;)&#9120;&#9126;"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "riem.Riemann_cont.getNonZero()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, `.Riemann_par` computes $R^{ab}_{\\ \\ cd}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAAbCAYAAADlJ3ZtAAAABHNCSVQICAgIfAhkiAAAAmpJREFU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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$R^{ab}_{\\ \\ \\ cd}$$"
      ],
      "text/plain": [
       "R_{\\ \\ \\ cd}__{ab}"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "riem.Riemann_par.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA/EAAAAhCAYAAACRFysnAAAABHNCSVQICAgIfAhkiAAACu1JREFU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\nC0aJhRfmDeYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( 0, \\quad 1, \\quad -1, \\quad 0\\right ), \\quad - \\frac{1}{a^{2}}\\right ), \\quad \\left ( \\left ( 0, \\quad 1, \\quad 0, \\quad -1\\right ), \\quad \\frac{1}{a^{2}}\\right ), \\quad \\left ( \\left ( 1, \\quad 0, \\quad -1, \\quad 0\\right ), \\quad \\frac{1}{a^{2}}\\right ), \\quad \\left ( \\left ( 1, \\quad 0, \\quad 0, \\quad -1\\right ), \\quad - \\frac{1}{a^{2}}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;               -1 &#9118;  &#9115;               1 &#9118;  &#9115;               1 &#9118;  &#9115;            \n",
       "&#9122;&#9116;(0, 1, -1, 0), &#9472;&#9472;&#9472;&#9119;, &#9116;(0, 1, 0, -1), &#9472;&#9472;&#9119;, &#9116;(1, 0, -1, 0), &#9472;&#9472;&#9119;, &#9116;(1, 0, 0, -1\n",
       "&#9122;&#9116;                 2&#9119;  &#9116;                2&#9119;  &#9116;                2&#9119;  &#9116;            \n",
       "&#9123;&#9117;                a &#9120;  &#9117;               a &#9120;  &#9117;               a &#9120;  &#9117;            \n",
       "\n",
       "   -1 &#9118;&#9124;\n",
       "), &#9472;&#9472;&#9472;&#9119;&#9125;\n",
       "     2&#9119;&#9125;\n",
       "    a &#9120;&#9126;"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "riem.Riemann_par.getNonZero()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Ricci tensor of a given curvature tensor is computed using `Ricci()`, with the curvature tensor as argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAWCAYAAAA4oUfxAAAABHNCSVQICAgIfAhkiAAAAcFJREFU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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$R_{ab}$$"
      ],
      "text/plain": [
       "R_{ab}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ricc = Ricci(riem)\n",
    "ricc.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWAAAAAaCAYAAACJr4ZfAAAABHNCSVQICAgIfAhkiAAACDNJREFU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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( -1, \\quad -1\\right ), \\quad \\sin^{2}{\\left (\\theta \\right )}\\right ), \\quad \\left ( \\left ( 0, \\quad 0\\right ), \\quad 1\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;             2   &#9118;             &#9124;\n",
       "&#9123;&#9117;(-1, -1), sin (&theta;)&#9120;, ((0, 0), 1)&#9126;"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ricc.getNonZero()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The curvature scalar $R=g^{ab}R_{ab}$ can be computed using `Scalar(Metric, Ricci, Riemann)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAAhCAYAAAA/F0BXAAAABHNCSVQICAgIfAhkiAAAAVFJREFU\nSInt1L9KXEEUgPHfmogasitWAYMkrKVhRSwEBQm4prCwsDBIuhQ2KfICkiaFsOAb2FnbamGl+AoB\ni6RRIU0gEkKEQLIWM5fcLHKdu26nHwycYe588+ecO/SASsHYDGYxjDl8xFEZ+WNs5vqr+IWnZSQN\n/MV47NfQjrJkKsJxsuNORMlUGUknO9i6jeAtWoqTUMhSlMAgnl/30YMCwTyeYU/I1su4m/PUHdTx\nQ7jMfKulCu46We7bvZDcRB/eYSj2W90stoyxGO9iunOFFOpYi/GXnLAUA6jGeB+j+cGiss/zB7+F\nF64dRV1Rw0a3kzPW0R9bsxvBa+GH/IbveJEfzNfJQ3zAmVAPr/BeyEYy23ESPMGl8BAl08BP/ypy\nEQepk7Nia+I4rg4LUTJSRnKBrzGuYgWHeJMiyYrtRHiUH2ESp0Kpf8LnFNE9/3MFX/MyfeaDtNkA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{2}{a^{2}}$$"
      ],
      "text/plain": [
       "2 \n",
       "&#9472;&#9472;\n",
       " 2\n",
       "a "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scurv = Scurv(gab,ricc)\n",
    "scurv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Einstein tensor $G_{ab}=R_{ab}-g_{ab}R/2$ can be computed using `Einstein(metric, Ricci)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAYCAYAAAAs7gcTAAAABHNCSVQICAgIfAhkiAAAAFhJREFU\nOI3t07EJgFAQg+FPsRc3cBErW3E0Z3ADQRzJ2gkULOwOLF5hIy9dwn9HmpCgIvgJHS5sGNCgxx6P\nZ7RvWZlSI8MZ/hNcBX9g8cxqxYgaZ8rTj3UDfaEKTXWjChAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\right ]$$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "einstein = Einstein(gab, ricc)\n",
    "einstein.getNonZero()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Verifying:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following property of spaces with constant curvature:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ R_{abcd}=\\alpha (g_{ad}g_{bc}-g_{bd}g_{ac}),$$\n",
    "$$ R_{ab}= \\alpha(d-1)g_{ab},$$\n",
    "\n",
    "Then \n",
    "$$R = \\alpha d(d-1)$$\n",
    "\n",
    "In our example $d=2$ and $R=2/a^2$, and therefore $\\alpha =1/a^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "alpha = 1/(a**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(0,0,0,0) True\n",
      "(0,0,0,1) True\n",
      "(0,0,1,0) True\n",
      "(0,0,1,1) True\n",
      "(0,1,0,0) True\n",
      "(0,1,0,1) True\n",
      "(0,1,1,0) True\n",
      "(0,1,1,1) True\n",
      "(1,0,0,0) True\n",
      "(1,0,0,1) True\n",
      "(1,0,1,0) True\n",
      "(1,0,1,1) True\n",
      "(1,1,0,0) True\n",
      "(1,1,0,1) True\n",
      "(1,1,1,0) True\n",
      "(1,1,1,1) True\n"
     ]
    }
   ],
   "source": [
    "for i in range(2):\n",
    "    for j in range(2):\n",
    "        for m in range(2):\n",
    "            for l in range(2):\n",
    "                cosa = gab.components[-i,-m]*gab.components[-j,-l]-gab.components[-j,-m]*gab.components[-i,-l]\n",
    "                print '(%d,%d,%d,%d)'%(i,j,m,l), cancel(riem.components[-i,-j,-m,-l])==simplify(alpha*cosa)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(0,0) True\n",
      "(0,1) True\n",
      "(1,0) True\n",
      "(1,1) True\n"
     ]
    }
   ],
   "source": [
    "for i in range(2):\n",
    "    for j in range(2):\n",
    "        print '(%d,%d)'%(i,j), cancel(gab.components[-i,-j])==simplify(ricc.components[-i,-j]/(alpha))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Killing Equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we will compute the Killing equation for the vector field $\\xi^a$, given by\n",
    "$$\n",
    "\\xi^c\\partial_c g_{ab}+g_{ac}\\partial_b\\xi^c+g_{bc}\\partial_a\\xi^c=0\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xi0 = Function('xi^0')(theta,phi)\n",
    "xi1 = Function('xi^1')(theta,phi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "xi = XI((theta,phi))\n",
    "xi[0] = xi0\n",
    "xi[1] = xi1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAWCAYAAABOm/V6AAAABHNCSVQICAgIfAhkiAAAAdJJREFU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\nvv/xb+MHtTNcd0mPEqMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$K_{ab}$$"
      ],
      "text/plain": [
       "K_{ab}"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = Killing_Equation(xi,gab)\n",
    "K.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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0itcI/23qeJOxUBVxENhjobpiARfSmteklvU5Psyi6/oF7bdRl/QLsu1RxqfXTdN62jUN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\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( -1, \\quad -1\\right ), \\quad a^{2} \\left(\\xi^{0}{\\left (\\theta,\\varphi \\right )} \\sin{\\left (2 \\theta \\right )} - \\cos{\\left (2 \\theta \\right )} \\frac{\\partial}{\\partial \\varphi} \\xi^{1}{\\left (\\theta,\\varphi \\right )} + \\frac{\\partial}{\\partial \\varphi} \\xi^{1}{\\left (\\theta,\\varphi \\right )}\\right)\\right ), \\quad \\left ( \\left ( -1, \\quad 0\\right ), \\quad a^{2} \\left(\\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta} \\xi^{1}{\\left (\\theta,\\varphi \\right )} + \\frac{\\partial}{\\partial \\varphi} \\xi^{0}{\\left (\\theta,\\varphi \\right )}\\right)\\right ), \\quad \\left ( \\left ( 0, \\quad -1\\right ), \\quad a^{2} \\left(\\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta} \\xi^{1}{\\left (\\theta,\\varphi \\right )} + \\frac{\\partial}{\\partial \\varphi} \\xi^{0}{\\left (\\theta,\\varphi \\right )}\\right)\\right ), \\quad \\left ( \\left ( 0, \\quad 0\\right ), \\quad 2 a^{2} \\frac{\\partial}{\\partial \\theta} \\xi^{0}{\\left (\\theta,\\varphi \\right )}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;           2 &#9115;                                     &part;                        \n",
       "&#9122;&#9116;(-1, -1), a &sdot;&#9116;&xi;&#8304;(&theta;, varphi)&sdot;sin(2&sdot;&theta;) - cos(2&sdot;&theta;)&sdot;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi)) + &#9472;&#9472;&#9472;\n",
       "&#9123;&#9117;             &#9117;                                  &part;varphi                  &part;va\n",
       "\n",
       "&part;                  &#9118;&#9118;  &#9115;          2 &#9115;   2    &part;                      &part;         \n",
       "&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi))&#9119;&#9119;, &#9116;(-1, 0), a &sdot;&#9116;sin (&theta;)&sdot;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi)) + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&#8304;(&theta;,\n",
       "rphi               &#9120;&#9120;  &#9117;            &#9117;        &part;&theta;                  &part;varphi      \n",
       "\n",
       "         &#9118;&#9118;  &#9115;          2 &#9115;   2    &part;                      &part;                  &#9118;\n",
       " varphi))&#9119;&#9119;, &#9116;(0, -1), a &sdot;&#9116;sin (&theta;)&sdot;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi)) + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&#8304;(&theta;, varphi))&#9119;\n",
       "         &#9120;&#9120;  &#9117;            &#9117;        &part;&theta;                  &part;varphi               &#9120;\n",
       "\n",
       "&#9118;  &#9115;           2 &part;                &#9118;&#9124;\n",
       "&#9119;, &#9116;(0, 0), 2&sdot;a &sdot;&#9472;&#9472;(&xi;&#8304;(&theta;, varphi))&#9119;&#9125;\n",
       "&#9120;  &#9117;             &part;&theta;               &#9120;&#9126;"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.getNonZero()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We asign the independent components of the equation to the variables e1, e2 and e3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "e1=K.components[(-1,-1)]\n",
    "e2=K.components[(-1,0)]\n",
    "e3=K.components[(0,0)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAAeCAYAAAAiu0AEAAAABHNCSVQICAgIfAhkiAAABmdJREFU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\nYII=\n",
      "text/latex": [
       "$$2 a^{2} \\frac{\\partial}{\\partial \\theta} \\xi^{0}{\\left (\\theta,\\varphi \\right )}$$"
      ],
      "text/plain": [
       "   2 &part;                \n",
       "2&sdot;a &sdot;&#9472;&#9472;(&xi;&#8304;(&theta;, varphi))\n",
       "     &part;&theta;               "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The 3rd equation has a solution $\\xi^0=f(\\varphi)$. We sunstitute this solution in the other equations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYgAAAAiCAYAAABFs04aAAAABHNCSVQICAgIfAhkiAAADRJJREFU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\n",
      "text/latex": [
       "$$a^{2} \\left(f{\\left (\\varphi \\right )} \\sin{\\left (2 \\theta \\right )} - \\cos{\\left (2 \\theta \\right )} \\frac{\\partial}{\\partial \\varphi} \\xi^{1}{\\left (\\theta,\\varphi \\right )} + \\frac{\\partial}{\\partial \\varphi} \\xi^{1}{\\left (\\theta,\\varphi \\right )}\\right)$$"
      ],
      "text/plain": [
       " 2 &#9115;                                 &part;                        &part;               \n",
       "a &sdot;&#9116;f(varphi)&sdot;sin(2&sdot;&theta;) - cos(2&sdot;&theta;)&sdot;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi)) + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varph\n",
       "   &#9117;                              &part;varphi                  &part;varphi            \n",
       "\n",
       "   &#9118;\n",
       "i))&#9119;\n",
       "   &#9120;"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Function('f')(phi)\n",
    "e1r=e1.subs(xi0,f)\n",
    "e1r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPYAAAAiCAYAAABsvSTxAAAABHNCSVQICAgIfAhkiAAAC21JREFU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\nkZGRkZGRkZGRkdEs/B9HzLIiAAot1wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$a^{2} \\left(\\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta} \\xi^{1}{\\left (\\theta,\\varphi \\right )} + \\frac{d}{d \\varphi} f{\\left (\\varphi \\right )}\\right)$$"
      ],
      "text/plain": [
       " 2 &#9115;   2    &part;                      d              &#9118;\n",
       "a &sdot;&#9116;sin (&theta;)&sdot;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi)) + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(f(varphi))&#9119;\n",
       "   &#9117;        &part;&theta;                  dvarphi           &#9120;"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2r=e2.subs(xi0,f)\n",
    "e2r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we solve for $f(\\varphi)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKcAAAAiCAYAAAA6a8mHAAAABHNCSVQICAgIfAhkiAAAB2xJREFU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\nHUlJSUlJSUlJSclhw/8BbHymGyx6MnIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ - \\tan{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\varphi} \\xi^{1}{\\left (\\theta,\\varphi \\right )}\\right ]$$"
      ],
      "text/plain": [
       "&#9121;           &part;                  &#9124;\n",
       "&#9122;-tan(&theta;)&sdot;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi))&#9125;\n",
       "&#9123;        &part;varphi               &#9126;"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sf=solve(e1r,f)\n",
    "sf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And substitute it back in equation 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAU8AAAAoCAYAAAB+S3ikAAAABHNCSVQICAgIfAhkiAAADGBJREFU\neJztnXmUHEUdxz+7m93chEAgJIFkSSIQE5Yj4oWGQFAJ8jjkChoxCZdcGgEFBDTwEARBIICCGhRR\n0WAeIKB4AJEIHgQR5BLhAYFgNAQQkKAi6x/fbqamp4/qa47d+rw3b6Z7qruqer7TVfWrX/0aHA6H\nw5Gajgbl2wn8EHgGeLZBZXA4+gPtwHHATGBn4K6GlqY1mABsC6xqdEGCtAHXAfMaXA6Hoz+wN7CF\n93kZML2BZWkVOoAfA3vEJWqvT1mq+BzwKvCdBuTtcPQ3JgKHeJ+foHIjdUTzP+ATwFk0UWOzAxqq\nj2x0QRrAJsC5wDnA1g0ui6NvEqaxgcBw7/PPgLENKFerMh14hMr1K4R3AZ8BFgG/BGZYHncH8Kki\nC5KCrGUuiqHAhsC7gSeR0ItkOBqWjS/4vI7WIU5jOwOnlJh3X9XfxcAlRZ1sGGrdfA4CXgPGJRz3\nIeAfwOCiCpKCrGUui4spthE5HDgT6AW6Czyvo3UxNbYBcHqJefVl/Y0BXga2KuJkPcCbwCRvewN0\n0Q5KOO4W4KIiCpCBrGUui09SYGtm0BfF68iGqbEjkYdLJ7B7iXn2Vf19E7iqiBO1oSFwm7c9FV20\nHWKO2QIZYXfKmOdM4AXghIzHpylzJ9EC6AauBM5Ds3HDUct0dEL+7cAZwO3A1cCNwNdtC+9xMrBx\nQhob8Y4H5kZ8101t/cqiA/25h5aYR7NRhrZ8ojR2MOo5PQ+8CEzLVPK+pz+w1+AMNFItfK7mGuDC\nhDTHAX/Pkcch6Ie5Msc5TKLK3AGcBgwJ+a4b+CMSNMCJwGLv82zie7FfBX4OdHnbTyOh2zIJuBW4\nHl2DMyPSJYl3BKp7Z8h33UTXryy2Ar5bch5RjEKNcr0oS1s+eTUWR1/VH9hpsA14DvXgC2MBcD6V\nHl0UtwBLc+Y1GRiQ8xwQX+Zjge1C9ncB93rH+hxItRPtdcCgkGMnAP8BtvG224F/Yf/HXegdfxVw\nKXABaozCSBLvpcCOIftt6lcWi4CP1yGfILO8vOtFGdryyauxOPq6/sBOg98Dbioqwz2pVHYQ0Ret\nDXgJzXQ3mrgyDwZ+FHHcZ1HL02XsOxyJxW9FP014D+Fg4DFjezbwOHYNwQFeHkss0kK8eCcCv4r4\nzqZ+ZTEKeID6r3a7gvrdPMvSlk8ejcXRH/QHdho8AjVIb13TrE7yM1D3+qfAZsgTf0xE2smou/5Q\nxrwGIp+19wHbZzwHJJf5A8A9IccNQvaeJagF9pnivfvXcAXhKxLWoUUBoOtwgXe+NyzKfBKytZxo\nkTaJY9GQKYht/crieWA1sGvJ+fh0AXOo7uWUTVna8smjsTj6g/7AToMPIpPLNjFp2A0Za89Hs0xz\n0Y/nMxEZoHsDrw0izrev9/2EmDz38/I7FfUGLgMu976bhmw5vVRWJe0H/BlV+gwkznOQ8fdu4D2B\n89uU+RJgl5CyzfPSTg3s/513Tp9Ngd+HHN+Ohl1L0Y17TkiaKB7w8kniY2hyoBfFDAgbVj1CxdvA\nZB529bPhSGoF2AZ8HvUsXkC9qCCnUx/7FmiF282ozo95n2+mMpwc6KU5C01c/ASYHzhHWv2VpS2f\nPBqLo9X0B+VpcIRXzqjJLg5Dguj2tsehrup9FoWO4ngv0yibzRQ0Q2gyh9rlmysD+zYGXkH21MOM\n/YuBpzKUcxm1PyBo1nI9Mpj7r9uQ94DZqHRSPXQqggWoRcw7/BqHeidh2NYviZGEi28xaow/goY+\nL1H7J9qL+ges6CV82H4ampX2lzGOAdYgHZuk0V8zasuGVtIflK/BVzEm4cyL0oPsQPOp/PirkZDu\nsCp6OL6T6esR3/eglnU4EiPIMBt0a3o1sL3Oe02k2gfrQST0TYC1Kco5DPh3YF8H6jFcD3zU2L8n\n6qGb12UI2VrKOFYCD6OWciQaQi0D/pLyPN1ogUKQNPVL4lDga4F9O6M/xgHGviHot33C2LcOeFuK\nvMpkrff6r7f9N/RnPgJNePik0V8zasuGVtIflK/BNcDm/oZ58zwH3aBMw/YE1GosTy53JMNR7zWK\nFejm+SwaIv0Bdf1t7Sx/Qr0IH99uMpR0N8/nqfXjGoe6678N7J/tvV9n7BtFseH1tga+jKLirM95\nrtHAP0P2p6lfEttS6/i/gFo/xaeotm2BhlIjEs6/HRp5JHl2+NxH7XDbhm8g+9ssdNNoQ73GKLOU\njf6aTVs2tJr+oHwNvozxO/o3zw2RQXoZlRYX5OrwJnBnwknjGExtq2vyHHJgPwW1OHOBryDhX2tx\n/rhzp+FR1KqYhv3R3vvDxr4BaOZzBbJ7+byDfD10k3ZkQ5pPfuGCyvxmyP409UsiOCs6DdmtgiJ9\nJ7JvB8uX9DveT/xCjKLYDuluJbKTPYsmGGdGpLfRXzNpy4ZW1B+Ur8HXMfx0/ZmsyagLHWwBdkEt\n60vAlgknjqI34fseVOijkJ1pPOr9XoGM9/XiVuD9gX3+bOUaY99s1FP+QiDtbsAN3ufgxFSaF8AH\n0TV5Ok+FDNYSvjoiTf2SCAY6mYNGEiY96DcN9kJGEj6sqyeXoJn4m9B1P5Tw3l4W80KzaMv21Yr6\ng/I1+AbGDdq/efonMp1SB6Khy6+97azuCuuJvwnuSPXs4DOoq92LesT14h508zZbL/96mC4fJyIv\nhOXGvjHI5OGLrS3HC9T7GZa/Sm+xmvCldbb1Aw3j4hy1n0GC9zU1jepVZbORF8XZIcdu4h1fT9ZT\nGXl1eZ+nIg3cEki7qfH5tAx5NYu2/FcHChpysvcKft+M+oPGa3AQRk/cz+Rx1MOc6G13eJkMQ0bV\nUcgmkIVXCF+SZnIS1a3G5sgobVa8k9pZv6h95nsaLkJBUH3WIdcT37drAWoIzIhI7cimErZkLWsM\nz4eQX+tVwEYpjovir2j2MhjL0aZ+INePR4Hvx+RxKrAPsjWeh8q9B3JbWY5EPQeNYoLsRLwrThnc\njWxkZv6r0Z+jx0jXg37HEVQ3cJBOf0VrKw97oZ7s+chkFgz422z6g+bQ4CA0aVbDZOTvdhGaVZyC\nZsBuQxdxdNhBFhyHepFdEd/PRa35hcCXUKtwGRVXkV1Qy92LbBe3owt0r7HvN6iFXopu8r0onmGW\nrv/eVLuVvB1dlyuQy0Ow5RtNdMCAuPiKQ5Hbwzx0cw2yBNXjNeDbyE6Th6vRSpQgSfXz06wl2QVs\nIAresgqV/WVkL59HvLPzndT6RpbNJCpBNBZTKd9MNNpaglyZFqJe00o0/J5Bdv0Vqa08LES+rKAb\n6L4haZpJf366RmtwDclxPAplH1SJ/h7+34yv2IaGhn6Lfy3h8QJnoR7CG+ga3oXhKpGS95J+9jLI\nIst0d1GZMU1iEroROdKR58kEttHlm01/0DgNtqPrUNdl5pPQhS8zjmArYMZXPAz1YnxuRJ4GUYwD\nvohs03nigN5A+CoPW85NTsJYNLO6aVJCj8tRoGxHOop4MoFtdPlm0R80ToP+fSzuf1oKL9K4x280\niqj4im3IxnygkXYVEnIYmxmfL0ZDrqxMBn6Ava+kya5UOxpHcTz2kz8zUJ0c+cjyZALb6PLNoj9o\nrAb3QjfPt65HESHebFiBWsh6rV9uBi5A9q09kF3saWQ3m44WH0xFrdlGqNdwf8R5tgc+jCYyhpA+\niLLJ42gFxlHIxmTLAOS+cqpF2mnYDc86kHCLCDrR33mUSiANW+agSZVONK8QFfGoGfQHjdfgdGRv\nXZOQrnCORjOZ/YW4+IrHUL2G9gRkkHc4bIl7MsFMKl4EILvlveiP7z/0sKjo8v2J5Wgiu+6MRcbW\nPCHlWom4+IqnUFkj3YbcQsKC5DocUcRFjTeXpE5BemtHs/Z5eo39mQ3R6qJZjSrADWjZZX9gd/RI\nAajEMt3f2z6QitF7AQp95nDYEjeqGU+17TwYeu3ksgvXRzkS+apmtdXmZjdkL4hbIdBXiIuvOAAF\nnziCSuAJh8OWuFHNjsjpHTSxYU5K7o7zeMnKPeSbKCuEXxD9/BOHw5FM3KimC/kMb4/+7KPRJM8x\nZFtW6lDj9CT1jbMRyrZeQZJCPzkcjnCSosaPQY+7eA15cNyGIt870jMALfXcPylhvVgIfKvRhXA4\n+jCjUXhJRz5OpwnvVdcQ8ywQh8ORi8Op72OV+yK7ovXugxtdkCAd6AaaN9iAw+Go5Uo0QevIxpbI\nFcyZFx0Oh8PhcDgcDofD4XA4HA6Hw+FwOBzR/B8XVKjRELnfyQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$a^{2} \\left(\\sin^{2}{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta} \\xi^{1}{\\left (\\theta,\\varphi \\right )} - \\tan{\\left (\\theta \\right )} \\frac{\\partial^{2}}{\\partial \\varphi^{2}}  \\xi^{1}{\\left (\\theta,\\varphi \\right )}\\right)$$"
      ],
      "text/plain": [
       "   &#9115;                                       2                  &#9118;\n",
       " 2 &#9116;   2    &part;                             &part;                   &#9119;\n",
       "a &sdot;&#9116;sin (&theta;)&sdot;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi)) - tan(&theta;)&sdot;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi))&#9119;\n",
       "   &#9116;        &part;&theta;                                2               &#9119;\n",
       "   &#9117;                                   &part;varphi                &#9120;"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2r2 = e2r.subs(f,sf[0]).doit()\n",
    "e2r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANQAAAAsCAYAAAAdH5ZsAAAABHNCSVQICAgIfAhkiAAACwJJREFU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\nPQ3ZZWD7hcVisVgsFovFYrG0OP4Pje+NKLSY9owAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\theta} \\xi^{1}{\\left (\\theta,\\varphi \\right )} - \\frac{\\frac{\\partial^{2}}{\\partial \\varphi^{2}}  \\xi^{1}{\\left (\\theta,\\varphi \\right )}}{\\cos{\\left (\\theta \\right )}}$$"
      ],
      "text/plain": [
       "                               2                  \n",
       "                              &part;                   \n",
       "                           &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi))\n",
       "                                  2               \n",
       "       &part;                   &part;varphi                \n",
       "sin(&theta;)&sdot;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi)) - &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "       &part;&theta;                           cos(&theta;)        "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2r3=trigsimp(e2r2/(a**2*sin(theta)))\n",
    "e2r3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#g, h = map(Function, 'gh')\n",
    "g = Function('g', real=True)(theta)\n",
    "h = Function('h', real=True)(phi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "sep=pde_separate(Eq(e2r3),xi1,[g,h],strategy='mul')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n = Symbol('n', real=True, integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAToAAAAeCAYAAAC8CNLIAAAABHNCSVQICAgIfAhkiAAACg5JREFU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\nHk+f4b9LB4YGPbKbrAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$g{\\left (\\theta \\right )} = C_{1} e^{\\frac{n^{2}}{2} \\left(- \\log{\\left (\\cos^{2}{\\left (\\theta \\right )} - 1 \\right )} + \\log{\\left (\\cos^{2}{\\left (\\theta \\right )} \\right )}\\right)}$$"
      ],
      "text/plain": [
       "            2 &#9115;     &#9115;   2       &#9118;      &#9115;   2   &#9118;&#9118;\n",
       "           n &sdot;&#9117;- log&#9117;cos (&theta;) - 1&#9120; + log&#9117;cos (&theta;)&#9120;&#9120;\n",
       "           &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "                             2                   \n",
       "g(&theta;) = C&#8321;&sdot;&#8495;                                      "
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sg=dsolve(sep[0]+n**2)\n",
    "sg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPsAAAAXCAYAAADN9DlfAAAABHNCSVQICAgIfAhkiAAAB7BJREFU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\n",
      "text/latex": [
       "$$h{\\left (\\varphi \\right )} = C_{1} \\sin{\\left (\\varphi \\left|{n}\\right| \\right )} + C_{2} \\cos{\\left (n \\varphi \\right )}$$"
      ],
      "text/plain": [
       "h(varphi) = C&#8321;&sdot;sin(varphi&sdot;&#9474;n&#9474;) + C&#8322;&sdot;cos(n&sdot;varphi)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sh=dsolve(sep[1]+n**2)\n",
    "sh"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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4HA6Hw+FwOBwOh8PhcDgcDofD4XA46pr/AYkICQCCE32/AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$C_{1} \\left(C_{1} \\sin{\\left (\\varphi \\left|{n}\\right| \\right )} + C_{2} \\cos{\\left (n \\varphi \\right )}\\right) e^{\\frac{n^{2}}{2} \\left(- \\log{\\left (\\cos^{2}{\\left (\\theta \\right )} - 1 \\right )} + \\log{\\left (\\cos^{2}{\\left (\\theta \\right )} \\right )}\\right)}$$"
      ],
      "text/plain": [
       "                                             2 &#9115;     &#9115;   2       &#9118;      &#9115;   2 \n",
       "                                            n &sdot;&#9117;- log&#9117;cos (&theta;) - 1&#9120; + log&#9117;cos (\n",
       "                                            &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "                                                              2               \n",
       "C&#8321;&sdot;(C&#8321;&sdot;sin(varphi&sdot;&#9474;n&#9474;) + C&#8322;&sdot;cos(n&sdot;varphi))&sdot;&#8495;                                  \n",
       "\n",
       "  &#9118;&#9118;\n",
       "&theta;)&#9120;&#9120;\n",
       "&#9472;&#9472;&#9472;&#9472;\n",
       "    \n",
       "    "
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sxi1 = sg.rhs*sh.rhs\n",
    "sxi1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAAiCAYAAABbc+vFAAAABHNCSVQICAgIfAhkiAAABrBJREFU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\nfoX6ljmMAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\tan{\\left (\\theta \\right )} \\frac{\\partial}{\\partial \\varphi} \\xi^{1}{\\left (\\theta,\\varphi \\right )}$$"
      ],
      "text/plain": [
       "           &part;                  \n",
       "-tan(&theta;)&sdot;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi))\n",
       "        &part;varphi               "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sf[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LTrcqmpE/pG/tXeR+F5MtZ1A7vso59qjnmHFV2rMdXzspKpNG2hXUtgnDm6idJJwVpWvH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\nCAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- C_{1} \\left(C_{1} \\cos{\\left (\\varphi \\left|{n}\\right| \\right )} \\left|{n}\\right| - C_{2} n \\sin{\\left (n \\varphi \\right )}\\right) e^{\\frac{n^{2}}{2} \\left(- \\log{\\left (\\cos^{2}{\\left (\\theta \\right )} - 1 \\right )} + \\log{\\left (\\cos^{2}{\\left (\\theta \\right )} \\right )}\\right)} \\tan{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "                                                    2 &#9115;     &#9115;   2       &#9118;     \n",
       "                                                   n &sdot;&#9117;- log&#9117;cos (&theta;) - 1&#9120; + lo\n",
       "                                                   &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "                                                                     2        \n",
       "-C&#8321;&sdot;(C&#8321;&sdot;cos(varphi&sdot;&#9474;n&#9474;)&sdot;&#9474;n&#9474; - C&#8322;&sdot;n&sdot;sin(n&sdot;varphi))&sdot;&#8495;                           \n",
       "\n",
       " &#9115;   2   &#9118;&#9118;       \n",
       "g&#9117;cos (&theta;)&#9120;&#9120;       \n",
       "&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;       \n",
       "                  \n",
       "           &sdot;tan(&theta;)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sf[0].subs(xi1,sxi1).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Checking Killing equations:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\xi^a_{1}(\\theta,\\varphi)=(\\sin(\\varphi),\\cot(\\theta)\\cos(\\theta)),$$\n",
    "\n",
    "$$\\xi^a_{2}(\\theta,\\varphi)=(\\cos(\\varphi),-\\cot(\\theta)\\sin(\\theta)),$$\n",
    "\n",
    "$$\\xi^a_{3}(\\theta,\\varphi)=(0,1),$$\n",
    "\n",
    "$$\n",
    "\\xi^a=\\alpha \\xi^a_{1} + \\beta \\xi^a_{2} + \\gamma \\xi^a_{3}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "alpha, beta, gamma = symbols('alpha, beta, gamma')\n",
    "xi1 = XI((theta,phi))\n",
    "xi[0] = alpha*sin(phi) + beta*cos(phi) \n",
    "xi[1] = alpha*cot(theta)*cos(phi) - beta*cot(theta)*sin(phi)+  gamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "K = Killing_Equation(xi,gab)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAYCAYAAAAs7gcTAAAABHNCSVQICAgIfAhkiAAAAFhJREFU\nOI3t07EJgFAQg+FPsRc3cBErW3E0Z3ADQRzJ2gkULOwOLF5hIy9dwn9HmpCgIvgJHS5sGNCgxx6P\nZ7RvWZlSI8MZ/hNcBX9g8cxqxYgaZ8rTj3UDfaEKTXWjChAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\right ]$$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.getNonZero()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "All the components of K_{ab} are 0!\n"
     ]
    }
   ],
   "source": [
    "K.__str__()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Covariant derivative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xi = XI((theta,phi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xi0 = Function('xi^0')(theta,phi)\n",
    "xi1 = Function('xi^1')(theta,phi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xi[0]= xi0\n",
    "xi[1]= xi1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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fVYn2uSOEt3WesWeZDONY2JU+Qfh1dqlPUK5OPYeWb9XNy8RHZo3QDn3KQ9v1yQjH7EwX\nT7dTRaBlVdjl0U18XGjdLMLDQoNshmbfJiTs5/HfwCEp962bJ2M+83IdvRDY/gzNcBDg9trMQeGV\nbyM51DiJqL61Jmq7nwa2e2vJL8/wHZug0Fc/C+mv3vIQvWueof8a3oomvh5FVUfuQCGjaddk/5xe\nh+t933S6fewpZPSiWBZVPbkIOMW3PWubrYJ+Rxh7AteS/OQpSpYgXye8BPLaKOdJsN1BkXFRuVBc\nMcx9OYxngddmkKcs0kRRmV/KR5P8Upsxm9YOmwZa9lJF7pIkphJts6LaOs/Ys0zaPhbOikt9gvDr\n7FKfoFydeh7dQE8hPi902fyF+KrkbdGnPLRZn5pEk3y42ZlePN1+MY0gWSe+PEWPu4GYTLr1svfQ\njKSHZfAEuhkLm5CYjm4Yg8ZyMtnXy5aFq2szEa0tPgQ3yhnVt2Z1/v4qsO9+aPIpLkFxkGCVinmo\nZHBQEbdGubuC8vmv4ePANqiqyR6oRPpnUXukSaaYpj/cjyYfotbNfwW4Afi3wPasbbYl0TP870b5\ny5KIkiXIQRHbxzvf81CK73LNsPflMJ5DOS7q5q9oUjUO80v5aJJfajNm09ph00CDaBft4YKoSf2o\nts4z9iyTto+Fs+BanyD8OrvUJyhXp7zIybSFNspiMXp4G0Vb9CkPbdWnptEUH252pp9V0KRXUmVH\nIPtTLe/mIu7kTxIfgTIMLEGRAGHrTTcgPH/ESgzeUpzdUGd92NH5ovqW51T9+TB2R9FWSZMsQYJV\nRPZH0Vp+NkW6EJwADl7DTdHvPxL1hbWB7wBfJvlGPS3XoVwgYYyhvFz+Njiw8zdrm+0C/EfI9hlo\n4i3JCMbJUiZLHbzA+nIYz9OMJUEvkJzU0vxSPurySy70tq5XmbiU0WxaONNILpRRBS8S/ZQ7qq3z\njD3LpG1j4SI65VqfIPw6u9QnKFenZqCb87LtYhJ/QJUdo2iLPuWhrfo0qD6cgrKZnelnFtErkvrI\nOvHlzbhNj9nnMZScu6nMIT7k1RXXoGgf/5KZ9ZEBDQsRXBV4JLCtKlnLYjNSlhdNSVTf8koN+58q\nnYCW1C0K7JvUpo8gpfZ0Yx566uOxO6rM+ImQY4PXcAuk3P5zL0TGa8UYGbJwJ+pTwdn096Df8PHA\n9h06f7O02Ww02R1maPdGa8DjSJKlTCY4eIH15TCmEr58oWq79Ufkk+Imv5rul6CZ9j6qH5Qtqwu9\nretVJi5lNJsWzixkU8KoUkefInriK86eZRl7VvF7XIyFoRpZi+iUa32C8OucRZ+gXp2aQbguVe3r\nfkl8jqA26VNW2qpPg+rDKSib2Zl+NqQ3Mi0Xi4CRkO3T0U37zjHHTkCGbo2iQuTkUiRj2E3Qzp3P\nspT6zstsFAb7Xt+2s1FkTFiSxdPRml2PKmUtiz3Rb/gqMNPB+eL61u10K8wtROuSg0qYpk1nojXZ\n9wCfRmVw56MSsIs6543q28FreDBa7uif6V4H+K8Q2b8R2PZYyLbDO/IHHfh2wGG+9zuiGfxvBF6X\n0rvEMk2bTQQ+RvSSthvolnwOI60saRgn3C6lYVWUUP90ZKCzMux9OYwf05+ssg67tVfnO9eN2adu\nvwTN8U1p5PEI6weD4JuysiHKUeGacfLZtKL2DMymRfEI4dWnqu73a6My7WE3YnFtnXbsWdXvKToW\nhurbPo9+udYniL7OafQJ6tepg+mvbF6H/9gGLeWKmjBokz75MR+enqb5cDA748rO3ElyFNoonaqO\nUSwi+kI+D/xTwhd8HXhnwj4uWQ24HlW988IBn0Q5iQ7w7ff6zvaHKpJrc/S04Cw0S3ol0cmgbwHe\n5HtftaxlcTG6Hi8CX0Prd4sQ1bdeD1yNlhGeS7hypm3TZYHj0Yz3UtTnb0FOPC5KMngN3wN8BPg8\n8Ek0k30+3bDpnZDCLkXrnG9CRuZnvm23Icd7GUrwtxR4kH4lfxuwcef/p4kOk/VHXaVps1lELxFb\nnfCZfz9pZUnDOPkdzHQUZbctar9g2G0ahrkvh3En8NGccrlkPfTb5ifsV7Vfgub5prTyeIT1gzp9\n0yTkV12zF92cp9OBU5GOnO7b52jcRep6jJPPprmwZ2A2Lcg0dIP8vgLyueQZovtHnD1LM/as8vcU\nGQtD9W0fpV9RtsHDtT5B+HVOo0/efnXq1BnomueRySVT0OTVW2L2aYs+td2HQzl+vG0+HMzOuLAz\n09G98o4J3z1KgYmve1Glgji2I3vG/yoZq1uAAOsTXflhrEI5ymJXlB/qFdTZbyd/NTgXfWss5X63\n061gkUTcNRxUjgOOqvD7xsnvYPycDXwg57HWl7v8nuiKhmMpv8sVzyCHGkfT/RI0y94n9YOxiuTw\nMw8NklyyO92HBhPQDc4bO++/jZ4Ug9rjGMffPU5xm1bEnoHZND9boDaIu5EYS/mdLrgOeEfEZ67s\n2ZiDc7iiiTbH06842+DHpT5Bu3XqOpQMu4hMrriM8EkED9On6nDtx9vuw8HsTBRJfXg+Gv9PSjjP\nKJ2Jrzwlm+8lOSTvJ2iGff0c568CV4nFXXE88OGIz5omax5uRFGC6yDFmAecmPNcLvpWmjZdA80w\np53MiruGg8q+aDBRFeeiJaBFuZ/8/cf6spiC2iCqqEHVdutW9NQsjqb7JWiWvU/qB1XLujLxNy55\n2RHlgAGF80+iqyvTUBJjgAeIzxOTBxc2rYg9A7NpfrZB+SzjklZX2e+vJTq1iCt7ZjYnHk+/4myD\nH5f6BO3VqclIn24oIJNLvgnsE/O56VM1lOHH2+7DwexMFEl92LsPfTXl+XJNfN2JMu8n8UG0lKmK\nRHFZ2JlmRebsiBJEXx/yWdNkLcLqyDichkIzHyxwriJ9K22b7o3kTVMNJe4aDioboNDxpyr8zrsJ\nT6aexEQUNnwTCufdHeVmGKV3zf+uqG88RHzYrPVlmIscTdjEVx126xq0dDiJpvolaJa9T+oHdci6\nAD3RXACcAmzp++wYNGA8HC0pn+I75j40MDwYlST/Ht0CPcvQDd+fAJyMlhF4vIHeCkNxhX3ykMem\nRdkzMJsWRhb/PIpyFwZLrGeVzxVXAnsQ3ZZF7ZnZnH6i9CvJNvhxqU/QTp16K4rseLaATC65Gt2Y\nbxezj+lT+UT58WHy4WB2xoWdmdY5XzDXcC4WER26tzFalzk54nM/OxCeK6EuJqM1501hElL8sBC9\npslalPnAecBngAsp/qQhT9/K0qYXAGem2C/uGg4y/4ZymLWBM5Hh9G4OH0YOx79Mby7wIeSMVgK+\nFHM+68tKSBnmjOqyW7OQg0yT66BpfgmaZe+T+kGdso7Tv0RiWZQE3Cuy8WXgSN/nn0ATox7fp7t8\nbDbdKkJboj40hmzBZ4C/0psQ+cICsrsiyp6B2bQgWWzaMmjJxHsjPq+r338X2CXm87z2zGxOOFH6\nlWQb/LjWJ2ifTl2KEnEXkck1R6DlYXGYPpXPOL1+fNh8OJidSSKNnTkeuCrl945SIMcXwO9ITiRm\nGMbg8gPcl9Qtg3XQ0/uNOu8nAkuAA4HtffsdGzjupPJFazVXUW1+tzRcjgYBxuAyTnhukHlIpw9F\nUTJjvs/G6C2k4T/Hcii3Bqg/3+7b73j0FNVjI3TjVCdR9mwUVQE0m5aft6NiLHEl2etgE+CHdQsx\nJETpV5JtMHpZHxVmahqTUFW5reoWZMgZp9+PD4sPB7MzLlgBTRaG5T8LY5QCOb5AVS3aEu1hGIZ7\nFqBcKE1nW7TM5/7O+39AZXvvR0khQeHCj/uOeQvNCVdvIjPRg49v1S1IgHOAdyOHaAw+3pKVdYEr\nUOj8xXSX3/qj0qPyP7xE1w68Bi1bAIX6H0p3QA2yHd8sJnJhouzZbcAqmE0rwpHoqflLdQsS4F7g\nN6RPBmzkJ0q/ZhJvG4xeziC52EwdvIoiOs+nu5TOqJftGC4fDmZnXPBZ1Da/zXpg3omvr6C8AzNz\nHm8YhlEFT9OdoJsBfA5FPvwcVaXcHNgflcudhp64bIPyvBjhHIYmvcJyd9TJbZ3X++sWxCiN59Gy\nPdBSPlCkzn2oghjAWp2/YaXdw7gD5Sx8gK6tOAQ9db6n835DVC5+SS6p3RFlz15BbWA2LR/boGUy\nn65bkAhOQnZ31aQdjUJE6ddviLYNRi/vQpPsd9QtSAS/AL6Arq1RD0E/Pkw+HMzOFGVvYEXgLJcn\nXURyec6T6Q09NAzDaBoT0RK4y9Bykf19n80GLgFeRM7lRuSAjWimA78HVqtbkAheBzyCcn4Zg8dc\npM9H0S1fvhYq+b1v53UgGsMcgB7Q3dV57dHZ9js0CeRPcrwferp8IUqueyzNLIAQZ8/AbFoeJqBJ\nwqbl/QsyE+VpS5Nf18hHlH61wTY0gU0pp/JuGbyP3jxSRnUE/fgw+XAwO1OEzVHUXtaIzVEK5vgC\nJWS7jfTrKw3DMJrGLPRUxUjHucgpN5mFKCG0YQwjZtOy8X6SE14bhmEYhtFORimY4wuUmO0glJ3f\nRSUBwzCMqllAN5+AEc98lD/roroFSeCrqFzycXULYhg1YDYtPdug6IJDknY0DMMwDGMwWQQ8CSzu\nvOKqAW2GJV8zDKOdXEB8qXhDbIQmvNqyzGYCqvA4v25BDKNizKalYzZwE5Y3yzAMwzAGjXPozmM9\nQ8JSR8MwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMwDMMw\nDMMwDMMwDMMwDMMwDMMwDMP4f/4P58FVwn27nKsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\left ( \\left ( -1, \\quad 0\\right ), \\quad - \\frac{1}{2} \\xi^{1}{\\left (\\theta,\\varphi \\right )} \\sin{\\left (2 \\theta \\right )} + \\frac{\\partial}{\\partial \\varphi} \\xi^{0}{\\left (\\theta,\\varphi \\right )}\\right ), \\quad \\left ( \\left ( -1, \\quad 1\\right ), \\quad \\frac{\\xi^{0}{\\left (\\theta,\\varphi \\right )}}{\\tan{\\left (\\theta \\right )}} + \\frac{\\partial}{\\partial \\varphi} \\xi^{1}{\\left (\\theta,\\varphi \\right )}\\right ), \\quad \\left ( \\left ( 0, \\quad 0\\right ), \\quad \\frac{\\partial}{\\partial \\theta} \\xi^{0}{\\left (\\theta,\\varphi \\right )}\\right ), \\quad \\left ( \\left ( 0, \\quad 1\\right ), \\quad \\frac{\\xi^{1}{\\left (\\theta,\\varphi \\right )}}{\\tan{\\left (\\theta \\right )}} + \\frac{\\partial}{\\partial \\theta} \\xi^{1}{\\left (\\theta,\\varphi \\right )}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;           &xi;&sup1;(&theta;, varphi)&sdot;sin(2&sdot;&theta;)      &part;                  &#9118;  &#9115;         &xi;&#8304;(&theta;,\n",
       "&#9122;&#9116;(-1, 0), - &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472; + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&#8304;(&theta;, varphi))&#9119;, &#9116;(-1, 1), &#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "&#9123;&#9117;                     2              &part;varphi               &#9120;  &#9117;             t\n",
       "\n",
       " varphi)      &part;                  &#9118;  &#9115;        &part;                &#9118;  &#9115;        &xi;&sup1;(&theta;\n",
       "&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472; + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;(&xi;&sup1;(&theta;, varphi))&#9119;, &#9116;(0, 0), &#9472;&#9472;(&xi;&#8304;(&theta;, varphi))&#9119;, &#9116;(0, 1), &#9472;&#9472;&#9472;&#9472;\n",
       "an(&theta;)      &part;varphi               &#9120;  &#9117;        &part;&theta;               &#9120;  &#9117;            \n",
       "\n",
       ", varphi)   &part;                &#9118;&#9124;\n",
       "&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472; + &#9472;&#9472;(&xi;&sup1;(&theta;, varphi))&#9119;&#9125;\n",
       "tan(&theta;)      &part;&theta;               &#9120;&#9126;"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Cov_der = Covariant_Derivative(xi, chris)\n",
    "Cov_der.getNonZero()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "======================================================================\n",
      "The non-zero components of nabla_{a} are:\n",
      "(-1, 0) : -xi^1(theta, varphi)*sin(2*theta)/2 + Derivative(xi^0(theta, varphi), varphi)\n",
      "(-1, 1) : (xi^0(theta, varphi) + tan(theta)*Derivative(xi^1(theta, varphi), varphi))/tan(theta)\n",
      "(0, 0) : Derivative(xi^0(theta, varphi), theta)\n",
      "(0, 1) : (xi^1(theta, varphi) + tan(theta)*Derivative(xi^1(theta, varphi), theta))/tan(theta)\n",
      "======================================================================\n"
     ]
    }
   ],
   "source": [
    "Cov_der.__str__()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Geodesic equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAZCAYAAABw43NsAAAABHNCSVQICAgIfAhkiAAAA/tJREFU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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$(Geoeq)^{a}$$"
      ],
      "text/plain": [
       "(Geoeq)__{a}"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Geoeq = Geodesic_Equation(chris)\n",
    "Geoeq.symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAhEAAAApCAYAAACLKHv7AAAABHNCSVQICAgIfAhkiAAAE1FJREFU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\nYII=\n",
      "text/latex": [
       "$$\\left [ \\left ( 0, \\quad - \\frac{1}{2} \\sin{\\left (2 \\theta \\right )} \\frac{d}{d l} \\varphi{\\left (l \\right )}^{2} + \\frac{d^{2}}{d l^{2}}  \\theta{\\left (l \\right )}\\right ), \\quad \\left ( 1, \\quad \\frac{d^{2}}{d l^{2}}  \\varphi{\\left (l \\right )} + \\frac{2 \\frac{d}{d l} \\theta{\\left (l \\right )}}{\\tan{\\left (\\theta \\right )}} \\frac{d}{d l} \\varphi{\\left (l \\right )}\\right )\\right ]$$"
      ],
      "text/plain": [
       "&#9121;&#9115;                             2            &#9118;                                 \n",
       "&#9122;&#9116;              &#9115;d            &#9118;             &#9119;  &#9115;                      d       \n",
       "&#9122;&#9116;     sin(2&sdot;&theta;)&sdot;&#9116;&#9472;&#9472;(varphi(l))&#9119;      2      &#9119;  &#9116;     2              2&sdot;&#9472;&#9472;(&theta;(l))\n",
       "&#9122;&#9116;              &#9117;dl           &#9120;     d       &#9119;  &#9116;    d                 dl      \n",
       "&#9122;&#9116;0, - &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472; + &#9472;&#9472;&#9472;(&theta;(l))&#9119;, &#9116;1, &#9472;&#9472;&#9472;(varphi(l)) + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "&#9122;&#9116;                 2                 2      &#9119;  &#9116;     2                       t\n",
       "&#9123;&#9117;                                 dl       &#9120;  &#9117;   dl                         \n",
       "\n",
       "               &#9124;\n",
       " d            &#9118;&#9125;\n",
       "&sdot;&#9472;&#9472;(varphi(l))&#9119;&#9125;\n",
       " dl           &#9119;&#9125;\n",
       "&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9119;&#9125;\n",
       "an(&theta;)         &#9119;&#9125;\n",
       "              &#9120;&#9126;"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Geoeq.getNonZero()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "======================================================================\n",
      "The non-zero components of (Geoeq)^{a} are:\n",
      "0 : -sin(2*theta)*Derivative(varphi(l), l)**2/2 + Derivative(theta(l), l, l)\n",
      "1 : (tan(theta)*Derivative(varphi(l), l, l) + 2*Derivative(theta(l), l)*Derivative(varphi(l), l))/tan(theta)\n",
      "======================================================================\n"
     ]
    }
   ],
   "source": [
    "Geoeq.__str__()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANUAAAAlCAYAAADV0qyaAAAABHNCSVQICAgIfAhkiAAACGRJREFU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\n",
      "text/latex": [
       "$$- \\frac{1}{2} \\sin{\\left (2 \\theta \\right )} \\frac{d}{d l} \\varphi{\\left (l \\right )}^{2} + \\frac{d^{2}}{d l^{2}}  \\theta{\\left (l \\right )}$$"
      ],
      "text/plain": [
       "                          2            \n",
       "           &#9115;d            &#9118;             \n",
       "  sin(2&sdot;&theta;)&sdot;&#9116;&#9472;&#9472;(varphi(l))&#9119;      2      \n",
       "           &#9117;dl           &#9120;     d       \n",
       "- &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472; + &#9472;&#9472;&#9472;(&theta;(l))\n",
       "              2                 2      \n",
       "                              dl       "
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "geq0 = Geoeq.components[0]\n",
    "geq0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAApCAYAAAAiY1vyAAAABHNCSVQICAgIfAhkiAAAB9ZJREFU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\n6dKly8jwf5TP8y7g1efGAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{d^{2}}{d l^{2}}  \\varphi{\\left (l \\right )} + \\frac{2 \\frac{d}{d l} \\theta{\\left (l \\right )}}{\\tan{\\left (\\theta \\right )}} \\frac{d}{d l} \\varphi{\\left (l \\right )}$$"
      ],
      "text/plain": [
       "                   d        d            \n",
       "  2              2&sdot;&#9472;&#9472;(&theta;(l))&sdot;&#9472;&#9472;(varphi(l))\n",
       " d                 dl       dl           \n",
       "&#9472;&#9472;&#9472;(varphi(l)) + &#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;&#9472;\n",
       "  2                       tan(&theta;)         \n",
       "dl                                       "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "geq1 = Geoeq.components[1]\n",
    "geq1"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": "3.6.2"
  }
 },
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