EfieldDerivatives.ipynb

{
 "metadata": {
  "name": "EfieldDerivatives"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Load up the sympy python package to do symbolic mathematics "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import display\n",
      "\n",
      "from sympy.interactive import printing\n",
      "printing.init_printing()\n",
      "\n",
      "from __future__ import division\n",
      "import sympy as sym\n",
      "\n",
      "\n",
      "\n",
      "import matplotlib\n",
      "matplotlib.rc('font',**{'family':'serif'})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E0    = sym.symbols(r\"\\mathcal{E}_{0}\", cls=sym.Function, real=True)\n",
      "k     = sym.symbols(r\"k\", real=True)\n",
      "l     = sym.symbols(r\"\\ell\", real=True)\n",
      "om    = sym.symbols(r\"\\omega\", real=True)\n",
      "t     = sym.symbols(r\"t\", real=True)\n",
      "vphi  = sym.symbols(r\"\\varphi\", cls=sym.Function, real=True)\n",
      "alpha = sym.symbols(r\"\\alpha\", cls=sym.Function, real=True)\n",
      "phi   = sym.symbols(r\"\\phi\", cls=sym.Function, real=True)\n",
      "\n",
      "Efield1 = E0(l)*sym.exp(sym.I*(k*l - om*t + vphi(l)))* sym.cos(alpha(l)) \n",
      "Efield2 = E0(l)*sym.exp(sym.I*(k*l - om*t + vphi(l)))* sym.sin(alpha(l)) *sym.exp(sym.I*phi(l))\n",
      "\n",
      "display(Efield1)\n",
      "display(Efield2)\n",
      "                                         "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANwAAAAZCAYAAABaf0CGAAAABHNCSVQICAgIfAhkiAAACO9JREFU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\nXfwfttj76E0gknIAAAAASUVORK5CYII=\n",
       "text": [
        "                       \u2148\u22c5(\\ell\u22c5k - \\omega\u22c5t + \\varphi(\\ell))                  \n",
        "\\mathcal{E}_{0}(\\ell)\u22c5\u212f                                     \u22c5cos(\\alpha(\\ell))"
       ]
      },
      {
       "latex": [
        "$$\\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)} e^{\\mathbf{\\imath} \\operatorname{\\phi}{\\left (\\ell \\right )}} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAAAZCAYAAAAxHpu/AAAABHNCSVQICAgIfAhkiAAACcZJREFU\neJzt3HvwVVUVB/APCKKCqWgaoikqGiMoKkgYGYTapDFmpUVBoYaG9jBrtMwa1Hygkko6Ymr5yEdl\nZJY5Nr5Ge6sFYVlkvpAyTStJMV/0xzrHe+757Xvv715+wI+Z852587tn7X32XnvvtdZee619f1Qo\nYtjaZmAdw1bYaG0zUWHV0HdtM9CLsD32WdtMrGN4Bp9b20xUqNBTmIM+CfphmIUp2BJXYeEa4mln\njF9DfeU4BP0Kz8Xx74zhpfp7Y/oa4Osj2LBEK/NKa34rlFB5AoHdsAwrS/SdMAnzMQhP4W78ZA3x\ndSJGdrPuJ3ugvylYgley5/L4l2B/9C+88xvs1wN9t8JArCg8l3mle/xWaAP9MA234t9iAW7HG9cm\nU6sJJ2NUgn4j3obJ4rgA38WENcOWx7BdN+vO7oH+zi49p8a/C6aW6n1NKODqwnqYUaKVeaX7/FYo\noJEnMAJ3iYk8C8fgD3gn9m2zj12wAOcJYblKBJRS6K+2eGV8VFj4udn7m7XJR46+OBaXCVcWxuKP\npXrDBO+/FfPxaPbuHvglDsK5WC6EtF0cKz3/V4uxfj0rn6pnjM5k9V7FZNwvxrUv1s8+OVLjhz+L\n+Sri9xjTIV+p9cgxHicJGXxD9ndogtd2+e0OJuJZHN/Bu2U0k+ueaqeoH6R1JKk3KSEcLlze4/FV\n3Ilrs8+juKMNpjcR3sMCfDZr808ZrbyI/cSCP5VoZyI+Js56K/A0TtBZZPpgfA8D1HbZjfBqqd77\n8WMcKQQU9sLj4gy8EF8QR4nyu62wAx7Ca9lz8ax7s1ioRdn3s/CzNttv1OcD2fcRwpiNxZ7ivL05\n/luonxp/jk1Kz89imw75Sq0HHI4P4UwxV/PEBnJVgtd2+e0Ohoh12KWDd4toJtc91c5E9fpBWkeS\nelMOqhA79lJx1itikbDM/2qD8ROEsl9foM3HKWKxLi7QP4Mf4IVSG/1xiRhgfmZ/TQjFqfh8G/zA\nbSIAOAlHZbTUTj4m6+dcvJjRDhAKdDj+gVvwSOGdT2HHJn3/GteJc+o3MtogMT/vyZ4fFsb2GGEE\nUtgKx6kPZE7ABoXn5Tg9+z5MvaczMRsXocCPiHUdVKiTGn+OV0rPK3Q16t1Faj22ETIyXKx5Ps7H\nMC7Ba7v8dgfX4V41j6JTNJLrnmonpR+kdaTberMcf5WOlLeLJfhRgr5YvUexMb7foI2Z+EvGz3pi\nZ8wxH2/ugK9ZuFAEm/rjp6Xy9YSrXN4F7hJCOEnNGO7aQf9fKnyfJLykHF8WHtrDIhvRV+x8rTC7\nSdmegm94k5rBIYJ6eWAvdyUbjZ8wckeVaB/AJ7rBYyOU1+MwNXkYJDyVvJ88KDu38H67/K4pNJPr\nnmqnmX6Q1pE6Wuo4sEy4jpdh2/b5fR0bC0v+eKLsb8K1zrG/OGencBy+I6zcB0XwJ8cdGa1dTMeV\nYkcndvWNC+UfFoYwt7qjhYCOFML2IP6elU3qoP+RYvyjcI5wPbcWwa6HMBgvCddvmvodvhM8IOZx\ntJive4RLeKwwDrdl9e4VAb7U+HNv6QBxNCxiSFa/U5TXY1HWJhGvuDt7Pk4YjCKvOuC3FQYIgzIh\na6tTNJPrnmqnmX6Q1pGWerOPWrrsNbGbN3NxG2Fk1saZibIFWdmA7Hke3pGot2dWb4wQ2rml8h3w\nw8R7o4W7PUctGFlU8ouFME3Onj8ugp5F7C7c6VlC0cfjmqysjwjgzRLK2y4OFJ7E78Q8LcaTartp\nP9wgIuKTE++nMLtF+RDB8/MikHe7yLMX0UcEIvvpOn4i754Ksl2udXym2ZqU14MQ0rOyekeKS0nF\nI0CRV23ym+MQYXi/KI4fF2afkSIrthJXlOovxj+Fx7Y/zsAF+IWudzoayTW8RczHecKdv154fsXx\ntWqnlX6Q1pFGevM6dhYTcDk+nT13gn0yBk9JlF2dleVZggXSbvUZYsL74DRdg0+biEUpYobYSYYW\naCeIAFQjDFY7P6+rOLEbdbbUM+5pERsIQW6GGdpfkxxHd8ZWS4zQNcg9Tb3S31d6Jo5my0W85sgC\nfZ6u8YNGcn0wnhCGK8dpWX+5B3NBN9pppR+kdaSOVj4O7CcGPVMMcJ7wBDpBHjEvX8ChdnkjH/BA\n/C9Rb1/hus4UQbUnSuUvqA9IjcelIqi2LKPtLaxoM7fs2eyzLt+BmNONOlN0FYhVxVS1IGcKna4J\nYWBWtKjTKXYTRrHoId6oPutQzkAQV6WfEbvpNwv0B0R2oyhDKbkeJXb9k4RhzHGtOCJOFsbymRbt\n0Fo/6KojXWjF7MAe4iLM26XP8e3i6SZlA7O/ywt1U3n/0WJwA0T6p4zN1KdMZgs3+73CdVtPnN+n\n4rkW/J4vDF8zgV7XMVascU9hWyGsDzapM1vna7K7yPuvDtwjjMATuElkw67X/ZuXC9VvcC9lfweq\nyX5Krs/J6NeU6LkCj8Zb1ac5O9UPuupIIxpiUPMbNNQJBoqYwgWJsjvUpxpP1vV8upGY5DOa9LGX\nmtL2F6mgSzphtsJqQW9fk2GCt6VC1l5Uf7vwLl2PA4TbX6bPyNrYvkAry/UWQicuasDPSmEcykfT\nTvWDeh1J0vLjwDhhdRdJY5RwlVL3ChrheRH4SmUYdlL/I5xbhAdSxEHZ3yua9DFBLZe+udhlViVK\nXaFn0ZvXZKw4Sx8tZPTNIsp+iVrAelVRlusdsz7va/LOKHG+b9YO3dMP6nUkScuNQH5TKxWJHi4u\nKczV/qWLm4WBKUY6dxSTfkOBdr+IXuexgukZT0+W3u2nPmc+TkwQESB5TtpQjRAZgAprFr15TXZV\nv+svxRFip960h/ooy3Xugi9P1N0w6/smXS87daof1OtIM5qhIgiyUpwtpovAzTEi9dHpvfAh4sdH\nxZ+ani9+h1AOVuwtFoFa1uBbeF/2fbBIJeXZigNxaKmNs0VOuTgxk0UAZ1Vz7RU6Q29dkxnirF1M\n8W6n/qbsz/HtxLvLEvSZQn9SP7U+ovB8qwi4FzFOyPZC4Yn01TWP365+kNaRLrTiwowR54v8H2ss\nFhHLS3W1TO1gtDjjLBGR2MHiCuTSRN13i6uh+RXXoeKHNI+LCT5d7C5b411iEoroJ6LkQ0WgZX0R\nWLpSOktRYfWjt67JNKH0W4gMRO4BzBGR/3OFTrwsfrsxRWTPviLy8y8Lg3GoiHvtJwJujwq5PLXQ\nV1GuNxUp1VdFXKyPuLdxTdbfRWLnv1JswBq0Q2P9IK0jSb3p5Grw+uIXXSm8jP900GaFChXWIdwp\nrE6jTzmAUaFChV6MTjyBbTW+RvwSfqX2E9kKFSpUqFChQoUKFSpUqFChQu/E/wFPkFkxy2KRLQAA\nAABJRU5ErkJggg==\n",
       "text": [
        "                       \u2148\u22c5(\\ell\u22c5k - \\omega\u22c5t + \\varphi(\\ell))  \u2148\u22c5\\phi(\\ell)    \n",
        "\\mathcal{E}_{0}(\\ell)\u22c5\u212f                                     \u22c5\u212f            \u22c5sin\n",
        "\n",
        "              \n",
        "(\\alpha(\\ell))"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "d11 = sym.simplify(sym.diff( Efield1, l, 1) )\n",
      "d12 = sym.simplify(sym.diff( Efield1, l, 2) )\n",
      "display(Efield1)\n",
      "display(d11)\n",
      "display(d12)\n",
      "display(sym.simplify(d12 + k**2 * Efield1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANwAAAAZCAYAAABaf0CGAAAABHNCSVQICAgIfAhkiAAACO9JREFU\neJztm32UVkUdxz+77KK8pYJC64YKCMSJlUVe9kCo0AodX1DpVEaBB0LIlUzJ0mNZrZkaKq6QBSqK\ngBh1DEs0j+ULRzmUJSWhqZRWgiRpiqyG+VZ/fGe888wz9z4vXHdZvN9znvM885u5M3NnfjO/3+87\n80AGF/3auwMdDH2Aru3diY6EyvbuwF6EI4Cx7d2JDoZ/A+e3dycydEzMByoC8s8CTcBkoDewAnis\njfo0CBjTRm1ZTAGqnLT7/oOAgV750cD0tulax0dm4YSjgOeB/3nyI4EJwBKgO/Av4CHgl23UrwuB\noUWW/XIK7U0GtgBvm7T//luAiUC188zvgONTaPsDjypgGnAvsBPYDdwPHNKenXqfcDFQF5D/HPg4\n0IhcToCfAuPaplv8Azi8yLLNKbR3pZcOvf9gYKpX7hq0ODMUQJyFGwKsQ4P8feBs4AngE8CxJbYx\nGFgDtKCJWYGC7RCqiSbWxxlop11gnj+oxH5YVAJzgaXIHQIYBfzZK9cP9f0PaDz+bp4dDvwGOAm4\nGmgFOpXRj7mEx38letcfmPyppLPAG8m1lo3ARvRexwKdzcci9P4AT6PxcvEnYGQKfUwLSXqUVj2F\n9LFoHR0IvEikjBZfBf5WbCUGBwDbkKW0+AbwOLmTC7Ko3yHMeo1H1rUC+B5SxCtiyhbCFBSLrQQ+\nY2S/DpT7GnAVctX2N7JRph8zgFrT53KYzf7AJ510F+f354BzgTPRhBaL5gL5s53fQ9D7VQI9geuB\nGjS2FqH3t7jJS59iyu8NSNKjtOoZT2F9DOqoGxxbtABbkW/uYhMK4F8podMXoIW12pEtAS4BZgGL\nHfm5wB3Af7w6qpFCNBHFWO8Ci4DvUvpE34cGagIwx8hCFmqkaedq4A0jm4SUdSawA7gHbUIW5wAD\nEtp+BPgxioNuMLLuaHxONulnkTU5G7g7pp4+wHnkkjzjyF0YrcBl5nc/ci34ePNeAC+bd3jF9MUi\n9P4Wb3vp3eRvoO2FOD1Kq55i9bFoHW0FniHM2JWKLcDagHwz8ICT7gH8LKaO2cBfTH86IRfXYglw\nWBn9agKuA7qhAfyVl98JuVuDPfk6oAEtVrvxfKyM9r/p/J4AzHPS30K747PIElcCvYqoszkh72jU\nb4APEy1uEOFhSY8F5jvu/UEbyhxP9mngrCL6+H4jSY/SqqdUfcyRhWKI55HLsxToW15/AXV6IPBc\nIG87MMJJT0RxUQjnAT9Bu8npKJC3eMDISsV0YDmyVCBr1cPJ/zzadOzuVo8W51CkiE8C/zR5E8po\nfyh6/zrkttUAhyLS4q/IzXsTsaLTyHfpSsXjaBzr0Xg9jFyduWgh3mfK/R6RH6H3t17AJOA2r/4a\nU769kaRHadVTqj4W1NGxRBT5u8hKJblJcRhq6rgikLfG5O1n0ouA4wLljjblRiIFWeDl9wd+EXiu\nHrls84mIGndBLUZWrtGkz0SEkIthyCVrQotqDLDK5FWgGLAJLZRScSKykH9E47QZeIHISlQBt6NY\nsTHwfAjNBfJrUJ9fRyTH/SiedVGBSJoq8t8fdA7nEyagmC4pZqpHc34z8CO06/skSwNwK5rjhWhD\n9C3sFLQpXYTCkuvMxyJOjwA+inSiBbmEq5EHYeF6dGnqY5yOvodB6CjgJuArJl0OxprOXRLIW2ny\nLFu5hrBrdjnwEhqMS4GPePkHIGV1MQPFm7WO7ALg1IS+9iSKdzoqLiyiTG/Scblc7I+UOA5jgV0o\nbrRYgY48LCYj9tM9crLM6HAn7YYhIOt/i5OO06NTEXk3zJFdCjxKZLkXFlFPOfoYkr2H44ENlBcX\n+WhAi6o5kLfa5FnrcC/hc5z1KHCdQ27cYVGNJspiDPAWOjuyGI3IB3c3C+F89s0zRhezSOe8zsVM\ntBhCqEDz45M/Leg8E0TUvIRcNR8LiW71nI5cY9dT6U6uhQvpUR0idc7w5EOQDk5Cm/O3C9QDpetj\nnsyN4YajQZhNOO4qFS8m5HUz361O2dBxQz3wIeR63hXIPwjFORbNyFU7DbFrLaaOqV65EK4l38Xa\n1zAK3ZRJC33RfconY/IbkIe00ZPPQ1fGQOeZvYCnAs8/jazSCBR39kaWaiVihLuQe8MmpEdXGfkq\nT77NfNejjWhpgXps2VL0MU/mLrhlaME9EaioHOxAO0io493Q7RW74J4i3zx3NeUeQYfAIfQlmuxq\nFO/cAXwdUbHzkN++q4j+vkNE1e+rOIt8t2xPsBW4MyH/CPO9LaFMf/PtHzWAvBWQtdmOFvBq5J4u\nMu27t158PToYWbC1aH5dWN0bhhbQ9oR6oHR9DMrsgmswDW+KqagOsTGhc7s4vI5IgRDTeSS5F4Dv\nAY7xypxkvm9JaGMckbvSC/njewNblkGwSpx0WcKyvSGX3x6H7EDWuQL4EtKpwxBbeD0R+ebr0QDz\nzKMJ7deheMxFGvoYlNkFZ+/rhRixgchqLCC8CyXhbrSYXQZoABqw2x3ZRsSi2Uux002fXvCerSL3\nTKoBDQ4oDthFeFMYgpjIDG2LDci6+QwwwKfQnK9Fxw+hM78R5vn1iMRwrdlW4IuIST/QyHw9sq5c\nK/noYp69k/yD/TT0MUlGLfAacgHvMg0ch247bKD8e3I1yHV0/75xLXJb/ZsJo9EAQsReLkMTA2IS\nFxOxpicSXc2yuBLFKO6gNCI6ek/PsjKUh4lIoV2W+BBybxl9AS2sGkfWD22i9khiBoqt3GOYw8m/\nEeXqEYgAWeSVaTDtP4YsZCX5Z2Vp6GOezFXMkYj2tH/C3IwOOG8kfwcoBfWIct+CGKae6NrM1kDZ\nExBdbK8h1SJ/+Tm0GVyGJuFQdBdxmfd8FTp7q0UT2Bldvl1O/l9vMrQdRiNCaydyIf+L5ulVp8wx\n6MxvJ4q3uqLN2VLq09ACOxixjtayzSdfl1w9OhCRZ+8gQq0CnUOuQjr/Q2TRliPjElcPlKaPQR0t\n5/pWZ8TUhPAWuYOYIUOGPcSDaHXHffxgM0OGDAblWLi+xF/1ehP4LTL3GTJkyJAhQ4YMGTJkyJAh\nXfwfttj76E0gknIAAAAASUVORK5CYII=\n",
       "text": [
        "                       \u2148\u22c5(\\ell\u22c5k - \\omega\u22c5t + \\varphi(\\ell))                  \n",
        "\\mathcal{E}_{0}(\\ell)\u22c5\u212f                                     \u22c5cos(\\alpha(\\ell))"
       ]
      },
      {
       "latex": [
        "$$\\left(\\mathbf{\\imath} k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )}\\right) e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAwYAAAAgCAYAAACo229iAAAABHNCSVQICAgIfAhkiAAAEpBJREFU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\nhmEYhmEYhmEYhmEYhjFC+T9vOyJ0kAYMCAAAAABJRU5ErkJggg==\n",
       "text": [
        "\u239b                                                                             \n",
        "\u239c\u2148\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell)) - \\mathcal{E}_{0}(\\ell)\u22c5sin(\\alph\n",
        "\u239d                                                                             \n",
        "\n",
        "           d                                                               d  \n",
        "a(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) + \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\n",
        "         d\\ell                                                           d\\ell\n",
        "\n",
        "                                      d                         \u239e  \u2148\u22c5(\\ell\u22c5k -\n",
        "(\\varphi(\\ell)) + cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u239f\u22c5\u212f           \n",
        "                                    d\\ell                       \u23a0             \n",
        "\n",
        " \\omega\u22c5t + \\varphi(\\ell))\n",
        "                          \n",
        "                          "
       ]
      },
      {
       "latex": [
        "$$\\left(- k^{2} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} - 2 \\mathbf{\\imath} k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} - 2 k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} k \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} - 2 \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\alpha}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )}\\right)^{2} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)^{2} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\varphi}{\\left (\\ell \\right )} - 2 \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )}\\right) e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)}$$"
       ],
       "output_type": "display_data",
       "png": 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tSvNuzDaux36nvbDZ19dg+1VfXZBOUdnJKX9RndTlbvKf8jkTs7P4uz+Ale2b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\nQgghhBBCCCGEEEIIIYQQQgghhBDCQ/4/xf1UerwPnEoAAAAASUVORK5CYII=\n",
       "text": [
        "\u239b                                                                             \n",
        "\u239c   2                                                                         \n",
        "\u239c- k \u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell)) - 2\u22c5\u2148\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5si\n",
        "\u239c                                                                             \n",
        "\u239d                                                                             \n",
        "\n",
        "                                                                              \n",
        "                  d                                                           \n",
        "n(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) - 2\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\el\n",
        "                d\\ell                                                         \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "      d                                              d                        \n",
        "l))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + 2\u22c5\u2148\u22c5k\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell)\n",
        "    d\\ell                                          d\\ell                      \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                                                  d                   d       \n",
        ") - 2\u22c5\u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\var\n",
        "                                                d\\ell               d\\ell     \n",
        "                                                                              \n",
        "\n",
        "                                                        2                     \n",
        "                                                       d                      \n",
        "phi(\\ell)) - \\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) - \\m\n",
        "                                                          2                   \n",
        "                                                     d\\ell                    \n",
        "\n",
        "                                                         2                    \n",
        "                                        d                                     \n",
        "athcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))  - \\mathcal{E}_{0}(\\\n",
        "                                      d\\ell                                   \n",
        "                                                                              \n",
        "\n",
        "                                           2                                  \n",
        "                         d                                                    \n",
        "ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell))  + \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\al\n",
        "                       d\\ell                                                  \n",
        "                                                                              \n",
        "\n",
        "              2                                                               \n",
        "             d                                           d                   d\n",
        "pha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) - 2\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\n",
        "                2                                      d\\ell               d\\e\n",
        "           d\\ell                                                              \n",
        "\n",
        "                                                                              \n",
        "                                                    d                         \n",
        "\u2500\u2500(\\mathcal{E}_{0}(\\ell)) + 2\u22c5\u2148\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\n",
        "ll                                                d\\ell                       \n",
        "                                                                              \n",
        "\n",
        "                                             2                         \u239e      \n",
        "   d                                        d                          \u239f  \u2148\u22c5(\\\n",
        "\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u239f\u22c5\u212f    \n",
        " d\\ell                                         2                       \u239f      \n",
        "                                          d\\ell                        \u23a0      \n",
        "\n",
        "                                 \n",
        "ell\u22c5k - \\omega\u22c5t + \\varphi(\\ell))\n",
        "                                 \n",
        "                                 \n",
        "                                 "
       ]
      },
      {
       "latex": [
        "$$\\left(- 2 \\mathbf{\\imath} k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} - 2 k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} k \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} - 2 \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\alpha}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )}\\right)^{2} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)^{2} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\varphi}{\\left (\\ell \\right )} - 2 \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )}\\right) e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)}$$"
       ],
       "output_type": "display_data",
       "png": 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yIWkHqtfP/uT/aHgR5lMXYvf2TEwnCzKOFT3xQOyGrSg7cIpYG3PurL+8JY7F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\nYII=\n",
       "text": [
        "\u239b                                                                             \n",
        "\u239c                                                  d                          \n",
        "\u239c- 2\u22c5\u2148\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) - 2\u22c5k\u22c5\\ma\n",
        "\u239c                                                d\\ell                        \n",
        "\u239d                                                                             \n",
        "\n",
        "                                                                              \n",
        "                                       d                                      \n",
        "thcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + 2\u22c5\u2148\u22c5k\u22c5cos(\\alpha(\\\n",
        "                                     d\\ell                                    \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "        d                                                                     \n",
        "ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell)) - 2\u22c5\u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell\n",
        "      d\\ell                                                                   \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "     d                   d                                                    \n",
        "))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) - \\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha\n",
        "   d\\ell               d\\ell                                                  \n",
        "                                                                              \n",
        "\n",
        "           2                                                                  \n",
        "          d                                                              d    \n",
        "(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) - \\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\\n",
        "             2                                                         d\\ell  \n",
        "        d\\ell                                                                 \n",
        "\n",
        "            2                                                               2 \n",
        "                                                          d                   \n",
        "alpha(\\ell))  - \\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell))  \n",
        "                                                        d\\ell                 \n",
        "                                                                              \n",
        "\n",
        "                                               2                              \n",
        "                                              d                               \n",
        "+ \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) - 2\u22c5sin(\\alp\n",
        "                                                 2                            \n",
        "                                            d\\ell                             \n",
        "\n",
        "                                                                              \n",
        "            d                   d                                             \n",
        "ha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell)) + 2\u22c5\u2148\u22c5cos(\\alpha(\\e\n",
        "          d\\ell               d\\ell                                           \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "       d                            d                                        d\n",
        "ll))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\n",
        "     d\\ell                        d\\ell                                       \n",
        "                                                                           d\\e\n",
        "\n",
        "2                         \u239e                                       \n",
        "                          \u239f  \u2148\u22c5(\\ell\u22c5k - \\omega\u22c5t + \\varphi(\\ell))\n",
        "\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u239f\u22c5\u212f                                     \n",
        "  2                       \u239f                                       \n",
        "ll                        \u23a0                                       "
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "d21 = sym.simplify( sym.diff( Efield2, l, 1) )\n",
      "d22 = sym.simplify( sym.diff( Efield2, l, 2) )\n",
      "display(Efield2)\n",
      "display(d21)\n",
      "display(d22)\n",
      "display(sym.simplify(d22 + k**2 * Efield2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)} e^{\\mathbf{\\imath} \\operatorname{\\phi}{\\left (\\ell \\right )}} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAAAZCAYAAAAxHpu/AAAABHNCSVQICAgIfAhkiAAACcZJREFU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\nAABJRU5ErkJggg==\n",
       "text": [
        "                       \u2148\u22c5(\\ell\u22c5k - \\omega\u22c5t + \\varphi(\\ell))  \u2148\u22c5\\phi(\\ell)    \n",
        "\\mathcal{E}_{0}(\\ell)\u22c5\u212f                                     \u22c5\u212f            \u22c5sin\n",
        "\n",
        "              \n",
        "(\\alpha(\\ell))"
       ]
      },
      {
       "latex": [
        "$$\\left(\\mathbf{\\imath} k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} + \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )}\\right) e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\phi}{\\left (\\ell \\right )} + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)}$$"
       ],
       "output_type": "display_data",
       "png": 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v+xg0ZMeP1LcNclabaF4bzmcKGvqTdd3HtWhYX5BngLuc7b1QgAWXozyb4uavnEHjnI99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\nYII=\n",
       "text": [
        "\u239b                                                                             \n",
        "\u239c\u2148\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell)) + \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\al\n",
        "\u239d                                                                             \n",
        "\n",
        "             d                                                             d  \n",
        "pha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell)) + \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\n",
        "           d\\ell                                                         d\\ell\n",
        "\n",
        "                                                            d                 \n",
        "(\\varphi(\\ell)) + \\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) \n",
        "                                                          d\\ell               \n",
        "\n",
        "                      d                         \u239e  \u2148\u22c5(\\ell\u22c5k - \\omega\u22c5t + \\phi\n",
        "+ sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u239f\u22c5\u212f                           \n",
        "                    d\\ell                       \u23a0                             \n",
        "\n",
        "(\\ell) + \\varphi(\\ell))\n",
        "                       \n",
        "                       "
       ]
      },
      {
       "latex": [
        "$$\\left(- k^{2} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} - 2 k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} - 2 k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} k \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )}\\right)^{2} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )}\\right)^{2} - 2 \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)^{2} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\phi}{\\left (\\ell \\right )} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\varphi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\alpha}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} + 2 \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )}\\right) e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\phi}{\\left (\\ell \\right )} + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)}$$"
       ],
       "output_type": "display_data",
       "png": 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ZC6/HGsrXpJY9LfV33Xqo+lVfDPXrrpY/AOtj9ePq1OvtB3wldfsw4GfYJevz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\nRERERERG6f8BMPZpAoTow7EAAAAASUVORK5CYII=\n",
       "text": [
        "\u239b                                                                             \n",
        "\u239c   2                                                                         \n",
        "\u239c- k \u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell)) - 2\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\n",
        "\u239c                                                                             \n",
        "\u239d                                                                             \n",
        "\n",
        "                                                                              \n",
        "                d                                                             \n",
        "\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell)) - 2\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\n",
        "              d\\ell                                                           \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "  d                                                                    d      \n",
        "\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + 2\u22c5\u2148\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\al\n",
        "d\\ell                                                                d\\ell    \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                                       d                                      \n",
        "pha(\\ell)) + 2\u22c5\u2148\u22c5k\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell)) - \\mathcal{E\n",
        "                                     d\\ell                                    \n",
        "                                                                              \n",
        "\n",
        "                                                 2                            \n",
        "                                d                                             \n",
        "}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))  - \\mathcal{E}_{0}(\\ell)\u22c5sin\n",
        "                              d\\ell                                           \n",
        "                                                                              \n",
        "\n",
        "                                2                                             \n",
        "                 d                                                            \n",
        "(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell))  - 2\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\n",
        "               d\\ell                                                          \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "  d                 d                                                         \n",
        "\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) - \\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell\n",
        "d\\ell             d\\ell                                                       \n",
        "                                                                              \n",
        "\n",
        "                       2                                                2     \n",
        "     d                                                                 d      \n",
        "))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell))  + \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\p\n",
        "   d\\ell                                                                  2   \n",
        "                                                                     d\\ell    \n",
        "\n",
        "                                                         2                    \n",
        "                                                        d                     \n",
        "hi(\\ell)) + \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + \n",
        "                                                           2                  \n",
        "                                                      d\\ell                   \n",
        "\n",
        "                                                                              \n",
        "                                              d                   d           \n",
        "2\u22c5\u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\el\n",
        "                                            d\\ell               d\\ell         \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                                                    d                   d     \n",
        "l)) + 2\u22c5\u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\v\n",
        "                                                  d\\ell               d\\ell   \n",
        "                                                                              \n",
        "\n",
        "                                                          2                   \n",
        "                                                         d                    \n",
        "arphi(\\ell)) + \\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) + \n",
        "                                                            2                 \n",
        "                                                       d\\ell                  \n",
        "\n",
        "                                                                              \n",
        "                        d                            d                        \n",
        "2\u22c5\u2148\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell)) + 2\u22c5\u2148\u22c5sin\n",
        "                      d\\ell                        d\\ell                      \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                 d                            d                               \n",
        "(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + sin(\\alpha(\n",
        "               d\\ell                        d\\ell                             \n",
        "                                                                              \n",
        "\n",
        "          2                                                                   \n",
        "         d                                                   d                \n",
        "\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell)) + 2\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\n",
        "            2                                              d\\ell              \n",
        "       d\\ell                                                                  \n",
        "\n",
        "                             \u239e                                                \n",
        "   d                         \u239f  \u2148\u22c5(\\ell\u22c5k - \\omega\u22c5t + \\phi(\\ell) + \\varphi(\\e\n",
        "\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u239f\u22c5\u212f                                              \n",
        " d\\ell                       \u239f                                                \n",
        "                             \u23a0                                                \n",
        "\n",
        "    \n",
        "ll))\n",
        "    \n",
        "    \n",
        "    "
       ]
      },
      {
       "latex": [
        "$$\\left(- 2 k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} - 2 k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} k \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} k \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )}\\right)^{2} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )}\\right)^{2} - 2 \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} - \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\left(\\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)^{2} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\phi}{\\left (\\ell \\right )} + \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\varphi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\alpha}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\phi}{\\left (\\ell \\right )} + 2 \\mathbf{\\imath} \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\varphi}{\\left (\\ell \\right )} + \\sin{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial^{2}}{\\partial^{2} \\ell}  \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )} + 2 \\cos{\\left (\\operatorname{\\alpha}{\\left (\\ell \\right )} \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\alpha}{\\left (\\ell \\right )} \\frac{\\partial}{\\partial \\ell} \\operatorname{\\mathcal{E}\\_{0}}{\\left (\\ell \\right )}\\right) e^{\\mathbf{\\imath} \\left(\\ell k - \\omega t + \\operatorname{\\phi}{\\left (\\ell \\right )} + \\operatorname{\\varphi}{\\left (\\ell \\right )}\\right)}$$"
       ],
       "output_type": "display_data",
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HqOM9kb4o2yJ+0lhO+qZ6Ndw2UfykTGkZyGLq30goQq6/1J8xS4Fv\nAC8HLhq4LDIiG2Ib1u5DF0RERERERERERERERERECh0LXJx4fAxwaYX3rQ9cAezQRaFEBtIkD8qC\niIiIiAxBYzkREREZG/VvRKQvr8FOJhWpZb2S5/8CPBnbuFZ1XxwRERERERERERERERERaeh64Obo\n762A44CXVXjfEcA1wC3R492wA1dEQtYkD8qCiIiIiAxBYzkREREZG/VvRKQPBwF7AS8duiASniUV\nX7cdcApwGHBrd8URERERERERERERERERkYZWAOdiB6usxC6a+vmK790VOBS4FrgKuKSLAor0qGke\nlAURERER6ZvGciIiIjI26t+ISNd2BF4NzAN/HbYoIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIi\nIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIi\nIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIiIi\nIiIp/wPFqOpvskT9mQAAAABJRU5ErkJggg==\n",
       "text": [
        "\u239b                                                                             \n",
        "\u239c                                                d                            \n",
        "\u239c- 2\u22c5k\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell)) - 2\u22c5k\u22c5\\mathca\n",
        "\u239c                                              d\\ell                          \n",
        "\u239d                                                                             \n",
        "\n",
        "                                                                              \n",
        "                                   d                                          \n",
        "l{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + 2\u22c5\u2148\u22c5k\u22c5\\mathcal{E}_{0}(\n",
        "                                 d\\ell                                        \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                          d                                             d     \n",
        "\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) + 2\u22c5\u2148\u22c5k\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\m\n",
        "                        d\\ell                                         d\\ell   \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                                                                 d            \n",
        "athcal{E}_{0}(\\ell)) - \\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\e\n",
        "                                                               d\\ell          \n",
        "                                                                              \n",
        "\n",
        "    2                                                            2            \n",
        "                                                  d                           \n",
        "ll))  - \\mathcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell))  - 2\u22c5\\mathca\n",
        "                                                d\\ell                         \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                                   d                 d                        \n",
        "l{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) - \\mat\n",
        "                                 d\\ell             d\\ell                      \n",
        "                                                                              \n",
        "\n",
        "                                                        2                     \n",
        "                                      d                                       \n",
        "hcal{E}_{0}(\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell))  + \u2148\u22c5\\mathcal{E}_{0}(\n",
        "                                    d\\ell                                     \n",
        "                                                                              \n",
        "\n",
        "                           2                                                  \n",
        "                          d                                                   \n",
        "\\ell)\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell)) + \u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5sin(\\alph\n",
        "                             2                                                \n",
        "                        d\\ell                                                 \n",
        "\n",
        "            2                                                                 \n",
        "           d                                                                  \n",
        "a(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + 2\u22c5\u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\ell))\u22c5\u2500\n",
        "              2                                                              d\n",
        "         d\\ell                                                                \n",
        "\n",
        "                                                                              \n",
        " d                   d                                                        \n",
        "\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell)) + 2\u22c5\u2148\u22c5\\mathcal{E}_{0}(\\ell)\u22c5cos(\\alpha(\\e\n",
        "\\ell               d\\ell                                                      \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "       d                   d                                                  \n",
        "ll))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + \\mathcal{E}_{0}(\\ell)\u22c5cos(\\alp\n",
        "     d\\ell               d\\ell                                                \n",
        "                                                                              \n",
        "\n",
        "             2                                                                \n",
        "            d                                            d                    \n",
        "ha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell)) + 2\u22c5\u2148\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\\n",
        "               2                                       d\\ell                  \n",
        "          d\\ell                                                               \n",
        "\n",
        "                                                                              \n",
        "        d                                         d                           \n",
        "ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\phi(\\ell)) + 2\u22c5\u2148\u22c5sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u22c5\u2500\n",
        "      d\\ell                                     d\\ell                        d\n",
        "                                                                              \n",
        "\n",
        "                                           2                                  \n",
        " d                                        d                                   \n",
        "\u2500\u2500\u2500\u2500(\\varphi(\\ell)) + sin(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell)) + 2\u22c5cos(\n",
        "\\ell                                         2                                \n",
        "                                        d\\ell                                 \n",
        "\n",
        "                                                              \u239e               \n",
        "                d                   d                         \u239f  \u2148\u22c5(\\ell\u22c5k - \\\n",
        "\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\alpha(\\ell))\u22c5\u2500\u2500\u2500\u2500\u2500(\\mathcal{E}_{0}(\\ell))\u239f\u22c5\u212f             \n",
        "              d\\ell               d\\ell                       \u239f               \n",
        "                                                              \u23a0               \n",
        "\n",
        "                                     \n",
        "omega\u22c5t + \\phi(\\ell) + \\varphi(\\ell))\n",
        "                                     \n",
        "                                     \n",
        "                                     "
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "k     = sym.symbols(r\"k\", real=True)\n",
      "l     = sym.symbols(r\"\\ell\", real=True)\n",
      "om    = sym.symbols(r\"\\omega\", real=True)\n",
      "t     = sym.symbols(r\"t\", real=True)\n",
      "E0    = sym.symbols(r\"\\mathcal{E}_{0}\", real=True)\n",
      "vphi  = sym.symbols(r\"\\varphi\", real=True)\n",
      "alpha = sym.symbols(r\"\\alpha\", real=True)\n",
      "phi   = sym.symbols(r\"\\phi\", real=True)\n",
      "\n",
      "dE0    = sym.symbols(r\"\\mathcal{E'}_{0}\", real=True)\n",
      "dvphi  = sym.symbols(r\"\\varphi'\", real=True)\n",
      "dalpha = sym.symbols(r\"\\alpha'\", real=True)\n",
      "dphi   = sym.symbols(r\"\\phi'\", real=True)\n",
      "\n",
      "P1 = sym.symbols(r\"P_{1}\", real=True)\n",
      "P2 = sym.symbols(r\"P_{2}\", real=True)\n",
      "P3 = sym.symbols(r\"P_{3}\", real=True)\n",
      "P4 = sym.symbols(r\"P_{4}\", real=True)\n",
      "\n",
      "re1=-2*k*E0*sym.cos(alpha)*dvphi + 4*sym.pi*k**2 * P1\n",
      "im1=-2*k*E0*sym.sin(alpha)*dalpha + 2*k*sym.cos(alpha)*dE0 - 4*sym.pi*k**2 * P2\n",
      "\n",
      "re2=-2*k*E0*sym.sin(alpha)*dphi - 2*k*E0*sym.sin(alpha)*dvphi + 4*sym.pi*k**2 * P3\n",
      "im2=2*k*E0*sym.cos(alpha)*dalpha + 2*k*sym.sin(alpha)*dE0 - 4*sym.pi*k**2 * P4 \n",
      "\n",
      "display(re1)\n",
      "display(im1)\n",
      "display(re2)\n",
      "display(im2)\n",
      "\n",
      "equations = [ re1, im1, re2, im2 ] \n",
      "variables = [dE0, dvphi, dalpha, dphi]\n",
      "sol = sym.solve(equations,variables, set=True)\n",
      "\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$4 \\pi P_{{1}} k^{2} - 2 \\mathcal{E}_{{0}} \\varphi' k \\cos{\\left (\\alpha \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALQAAAAcCAYAAADIgpWxAAAABHNCSVQICAgIfAhkiAAAB9tJREFU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\niYmJiYmJiYmJicx/wk7TpsuarF0AAAAASUVORK5CYII=\n",
       "text": [
        "           2                                           \n",
        "4\u22c5\u03c0\u22c5P_{1}\u22c5k  - 2\u22c5\\mathcal{E}_{0}\u22c5\\varphi'\u22c5k\u22c5cos(\\alpha)"
       ]
      },
      {
       "latex": [
        "$$- 4 \\pi P_{{2}} k^{2} - 2 \\alpha' \\mathcal{E}_{{0}} k \\sin{\\left (\\alpha \\right )} + 2 \\mathcal{E'}_{{0}} k \\cos{\\left (\\alpha \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASkAAAAlCAYAAAAeLB9NAAAABHNCSVQICAgIfAhkiAAACghJREFU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\njAAbyE41dwAAAABJRU5ErkJggg==\n",
       "text": [
        "             2                                                                \n",
        "- 4\u22c5\u03c0\u22c5P_{2}\u22c5k  - 2\u22c5\\alpha'\u22c5\\mathcal{E}_{0}\u22c5k\u22c5sin(\\alpha) + 2\u22c5\\mathcal{E'}_{0}\u22c5\n",
        "\n",
        "             \n",
        "k\u22c5cos(\\alpha)"
       ]
      },
      {
       "latex": [
        "$$4 \\pi P_{{3}} k^{2} - 2 \\mathcal{E}_{{0}} \\phi' k \\sin{\\left (\\alpha \\right )} - 2 \\mathcal{E}_{{0}} \\varphi' k \\sin{\\left (\\alpha \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASIAAAAcCAYAAADFubLfAAAABHNCSVQICAgIfAhkiAAACZlJREFU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\n/wPhIx4nsWql7wAAAABJRU5ErkJggg==\n",
       "text": [
        "           2                                                                  \n",
        "4\u22c5\u03c0\u22c5P_{3}\u22c5k  - 2\u22c5\\mathcal{E}_{0}\u22c5\\phi'\u22c5k\u22c5sin(\\alpha) - 2\u22c5\\mathcal{E}_{0}\u22c5\\varp\n",
        "\n",
        "                 \n",
        "hi'\u22c5k\u22c5sin(\\alpha)"
       ]
      },
      {
       "latex": [
        "$$- 4 \\pi P_{{4}} k^{2} + 2 \\alpha' \\mathcal{E}_{{0}} k \\cos{\\left (\\alpha \\right )} + 2 \\mathcal{E'}_{{0}} k \\sin{\\left (\\alpha \\right )}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAS4AAAAlCAYAAAD88AQ0AAAABHNCSVQICAgIfAhkiAAAClVJREFU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\nAoFAIBAIBAKBQCAQCAQCgXbn/9ISBc5xWaVVAAAAAElFTkSuQmCC\n",
       "text": [
        "             2                                                                \n",
        "- 4\u22c5\u03c0\u22c5P_{4}\u22c5k  + 2\u22c5\\alpha'\u22c5\\mathcal{E}_{0}\u22c5k\u22c5cos(\\alpha) + 2\u22c5\\mathcal{E'}_{0}\u22c5\n",
        "\n",
        "             \n",
        "k\u22c5sin(\\alpha)"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Now need to solve the above 4 equation with 4 unknowns, the unknowns are the derivatives."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sol"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\begin{bmatrix}\\alpha', & \\mathcal{E'}_{{0}}, & \\phi', & \\varphi'\\end{bmatrix}, & \\left\\{\\begin{pmatrix}2 \\frac{\\pi k \\left(- P_{{2}} \\sin{\\left (\\alpha \\right )} + P_{{4}} \\cos{\\left (\\alpha \\right )}\\right)}{\\mathcal{E}_{{0}}}, & 2 \\pi k \\left(P_{{2}} \\cos{\\left (\\alpha \\right )} + P_{{4}} \\sin{\\left (\\alpha \\right )}\\right), & 2 \\frac{\\pi k \\left(- P_{{1}} \\sin{\\left (\\alpha \\right )} + P_{{3}} \\cos{\\left (\\alpha \\right )}\\right)}{\\mathcal{E}_{{0}} \\sin{\\left (\\alpha \\right )} \\cos{\\left (\\alpha \\right )}}, & 2 \\frac{\\pi P_{{1}} k}{\\mathcal{E}_{{0}} \\cos{\\left (\\alpha \\right )}}\\end{pmatrix}\\right\\}\\end{pmatrix}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 65,
       "text": [
        "\u239b                                                 \u239b\u23a1\u239b2\u22c5\u03c0\u22c5k\u22c5(-P_{2}\u22c5sin(\\alpha)\n",
        "\u239c[\\alpha', \\mathcal{E'}_{0}, \\phi', \\varphi'], set\u239c\u23a2\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u239d                                                 \u239d\u23a3\u239d               \\mathcal{E\n",
        "\n",
        " + P_{4}\u22c5cos(\\alpha))                                                 2\u22c5\u03c0\u22c5k\u22c5(-\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 2\u22c5\u03c0\u22c5k\u22c5(P_{2}\u22c5cos(\\alpha) + P_{4}\u22c5sin(\\alpha)), \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "}_{0}                                                                    \\math\n",
        "\n",
        "P_{1}\u22c5sin(\\alpha) + P_{3}\u22c5cos(\\alpha))          2\u22c5\u03c0\u22c5P_{1}\u22c5k        \u239e\u23a4\u239e\u239e\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u23a5\u239f\u239f\n",
        "cal{E}_{0}\u22c5sin(\\alpha)\u22c5cos(\\alpha)      \\mathcal{E}_{0}\u22c5cos(\\alpha)\u23a0\u23a6\u23a0\u23a0"
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sym.pi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\pi$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAANCAYAAACZ3F9/AAAABHNCSVQICAgIfAhkiAAAALJJREFU\nKJHN0DFqAlEYBOBvNYVYLBam8AZeQisRbFLYiWCXPuBdPICVkCKFpYWdCha5wtqkTZkgmKTIDy6y\nooWFA483b2D+mf9xA6SY4Qu/Z84PWpCEKcEb3rFBH1tkeMEEe3xjFQPACL1c+ivKwedF9R7inua0\nWugHVPFYZCwVaEOsgzdjwFXGZyyDN1AvMp6ijU/H/Qb+f7lyKXGMRa7eR5g6lxIzdHPvFDs8XVP3\nTvEHblUhqB0eKK0AAAAASUVORK5CYII=\n",
       "prompt_number": 63,
       "text": [
        "\u03c0"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}