EMsage.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Electromagnetism"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Running Sage Math and sagemanifolds with jupyter notebooks tip*: First, I've found that compiling from the github source the development Sage Math version (see my notes on this under the \"Computers\" section of my [wordpress blog](https://ernestyalumni.wordpress.com)) works with sagemanifolds and either installing sagemanifolds into the binary that you unpack out (the click, download, double click), \"breaks\" Sage and so that it doesn't run.  Also, following the Sage Math instructions on their website for building from the source didn't work for me (!!!???).  \n",
    "\n",
    "As a tip on how to run this jupyter notebook and have sagemanifolds available, you'd want to be in the working directory you desire (e.g. `Propulsion/EM`).  But yet your Sage Math build is somewhere else (e.g. `/home/topolo/Public/sage`).  Do this out of the working directory you're currently working out of:\n",
    "`/home/topolo/Public/sage/sage -n jupyter`\n",
    "\n",
    "For the rationale, or the math, and how the math corresponds directly to the Sage Math code here, you're going to want to look at [Gravity_Notes_grande.pdf](https://github.com/ernestyalumni/Gravite/blob/master/LaTeXetpdfs/Gravity_Notes_grande.pdf) in my Gravite repository, and in there, the $\\mathbb{R}^3$ section, because I define the charts and atlases for Euclidean space $\\mathbb{R}^3$ as a *smooth manifold*.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%display latex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = Manifold(4,'M',r'M')\n",
    "cart_ch = M.chart('t x y z')\n",
    "U = M.open_subset('U',coord_def={cart_ch: (cart_ch[1]<0, cart_ch[2]!=0)})\n",
    "cart_ch_U = cart_ch.restrict(U)\n",
    "sph_ch = U.chart(r't:(-oo,oo):t rh:(0,+oo):\\rho th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n",
    "t, rh,th,ph = [sph_ch[i[0]] for i in M.index_generator(1)]\n",
    "transit_sph_to_cart = sph_ch.transition_map(cart_ch_U, \n",
    "                                            [t,rh*sin(th)*cos(ph),rh*sin(th)*sin(ph),rh*cos(th)])\n",
    "Sphnorm = sqrt(sum([cart_ch_U[i]**2 for i in range(1,4)]))\n",
    "transit_sph_to_cart.set_inverse( cart_ch[0], Sphnorm, \n",
    "                                atan2(sqrt( sum([ cart_ch_U[i]**2 for i in range(1,3)])),cart_ch_U[3]),\n",
    "                                                            atan2(cart_ch_U[2],cart_ch_U[1]))\n",
    "cyl_ch = U.chart(r't:(-oo,oo):t r:(0,+oo) phi:(0,2*pi):\\phi z:(-oo,oo):z')\n",
    "t, r,phi,z = [cyl_ch[i[0]] for i in M.index_generator(1)]\n",
    "transit_cyl_to_cart = cyl_ch.transition_map(cart_ch_U, [t,r*cos(phi),r*sin(phi),z])\n",
    "transit_cyl_to_cart.set_inverse(cart_ch_U[0], sqrt(cart_ch_U[1]**2+cart_ch_U[2]**2), \n",
    "                                    atan2( cart_ch_U[2],cart_ch_U[1]), cart_ch_U[3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note the mostly positive (-+++) convention I use for the Minkowski metric."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "g = M.lorentzian_metric('g')\n",
    "g[0,0] = -1 \n",
    "for i in range(1,4): g[i,i] = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Electric Field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_E(ch):\n",
    "    \"\"\"\n",
    "    make_E = make_E(ch)\n",
    "    make_E creates a time-INDEPENDENT electric field as a 1-form\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    Ecomplst = []\n",
    "    for i in range(1,4):\n",
    "        Earglst = ['E'+str(i),] + list(ch[1:])\n",
    "        Ecomplst.append( function(Earglst[0])(*Earglst[1:]) )\n",
    "    Ecomplst = [0,]+Ecomplst\n",
    "    E = ch.domain().diff_form(1)\n",
    "    E[ch.frame(),:,ch] = Ecomplst\n",
    "    return E\n",
    "\n",
    "def make_Et(ch):\n",
    "    \"\"\"\n",
    "    make_Et = make_Et(ch)\n",
    "    make_Et creates a time-DEPENDENT electric field as a 1-form\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    Ecomplst = []\n",
    "    for i in range(1,4):\n",
    "        Earglst = ['E'+str(i),] + list(ch[:])\n",
    "        Ecomplst.append( function(Earglst[0])(*Earglst[1:]) )\n",
    "    Ecomplst = [0,]+Ecomplst\n",
    "    E = ch.domain().diff_form(1)\n",
    "    E[ch.frame(),:,ch] = Ecomplst\n",
    "    return E"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Examples of using `make_E`, `make_Et` and displaying the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E1(x, y, z) dx + E2(x, y, z) dy + E3(x, y, z) dz\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}E_{1}\\left({t}, {\\rho}, {\\theta}, {\\phi}\\right) \\mathrm{d} {\\rho} + E_{2}\\left({t}, {\\rho}, {\\theta}, {\\phi}\\right) \\mathrm{d} {\\theta} + E_{3}\\left({t}, {\\rho}, {\\theta}, {\\phi}\\right) \\mathrm{d} {\\phi}</script></html>"
      ],
      "text/plain": [
       "E1(t, rh, th, ph) drh + E2(t, rh, th, ph) dth + E3(t, rh, th, ph) dph"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print make_E(cart_ch).display()\n",
    "make_Et(sph_ch).display(sph_ch.frame(),sph_ch)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Magnetic Field"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Programming note: `make_B` and `make_Bt` "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_B(ch):\n",
    "    \"\"\"\n",
    "    make_B = make_B(ch)\n",
    "    make_B creates a time-INDEPENDENT magnetic field as a 2-form\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    B = ch.domain().diff_form(2)\n",
    "    farglst = list(ch[1:]) # function argument list, e.g. (x,y,z)\n",
    "\n",
    "    B[ch.frame(),1,2,ch] = function('B_12')(*farglst)\n",
    "    B[ch.frame(),2,3,ch] = function('B_23')(*farglst)\n",
    "    B[ch.frame(),3,1,ch] = function('B_31')(*farglst)\n",
    "\n",
    "    return B\n",
    "\n",
    "def make_Bt(ch):\n",
    "    \"\"\"\n",
    "    make_Bt = make_Bt(ch)\n",
    "    make_Bt creates a time-DEPENDENT electric field as a 2-form\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    B = ch.domain().diff_form(2)\n",
    "    farglst = list(ch[:]) # function argument list, e.g. (x,y,z)\n",
    "\n",
    "    B[ch.frame(),1,2,ch] = function('B_12')(*farglst)\n",
    "    B[ch.frame(),2,3,ch] = function('B_23')(*farglst)\n",
    "    B[ch.frame(),3,1,ch] = function('B_31')(*farglst)\n",
    "\n",
    "    return B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "B_12(t, x, y, z) dx/\\dy - B_31(t, x, y, z) dx/\\dz + B_23(t, x, y, z) dy/\\dz\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}B_{12}\\left(r, {\\phi}, {z}\\right) \\mathrm{d} r\\wedge \\mathrm{d} {\\phi} -B_{31}\\left(r, {\\phi}, {z}\\right) \\mathrm{d} r\\wedge \\mathrm{d} {z} + B_{23}\\left(r, {\\phi}, {z}\\right) \\mathrm{d} {\\phi}\\wedge \\mathrm{d} {z}</script></html>"
      ],
      "text/plain": [
       "B_12(r, phi, z) dr/\\dphi - B_31(r, phi, z) dr/\\dz + B_23(r, phi, z) dphi/\\dz"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print make_Bt(cart_ch).display()\n",
    "make_B(cyl_ch).display(cyl_ch.frame(),cyl_ch)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the orientation is correct (with the right hand rule)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Electromagnetic field 2-form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-E_{1}\\left({t}, x, y, {z}\\right) \\mathrm{d} t\\wedge \\mathrm{d} x -E_{2}\\left({t}, x, y, {z}\\right) \\mathrm{d} t\\wedge \\mathrm{d} y -E_{3}\\left({t}, x, y, {z}\\right) \\mathrm{d} t\\wedge \\mathrm{d} z + B_{12}\\left({t}, x, y, {z}\\right) \\mathrm{d} x\\wedge \\mathrm{d} y -B_{31}\\left({t}, x, y, {z}\\right) \\mathrm{d} x\\wedge \\mathrm{d} z + B_{23}\\left({t}, x, y, {z}\\right) \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "-E1(t, x, y, z) dt/\\dx - E2(t, x, y, z) dt/\\dy - E3(t, x, y, z) dt/\\dz + B_12(t, x, y, z) dx/\\dy - B_31(t, x, y, z) dx/\\dz + B_23(t, x, y, z) dy/\\dz"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EM_F = make_Bt(cart_ch) + make_Et(cart_ch).wedge(cart_ch.coframe()[0] )\n",
    "EM_F.display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "0 & -E_{1}\\left({t}, x, y, {z}\\right) & -E_{2}\\left({t}, x, y, {z}\\right) & -E_{3}\\left({t}, x, y, {z}\\right) \\\\\n",
       "E_{1}\\left({t}, x, y, {z}\\right) & 0 & B_{12}\\left({t}, x, y, {z}\\right) & -B_{31}\\left({t}, x, y, {z}\\right) \\\\\n",
       "E_{2}\\left({t}, x, y, {z}\\right) & -B_{12}\\left({t}, x, y, {z}\\right) & 0 & B_{23}\\left({t}, x, y, {z}\\right) \\\\\n",
       "E_{3}\\left({t}, x, y, {z}\\right) & B_{31}\\left({t}, x, y, {z}\\right) & -B_{23}\\left({t}, x, y, {z}\\right) & 0\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[                0   -E1(t, x, y, z)   -E2(t, x, y, z)   -E3(t, x, y, z)]\n",
       "[   E1(t, x, y, z)                 0  B_12(t, x, y, z) -B_31(t, x, y, z)]\n",
       "[   E2(t, x, y, z) -B_12(t, x, y, z)                 0  B_23(t, x, y, z)]\n",
       "[   E3(t, x, y, z)  B_31(t, x, y, z) -B_23(t, x, y, z)                 0]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EM_F[:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, \\frac{\\partial\\,B_{12}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial y} + \\frac{\\partial\\,E_{2}}{\\partial x}, -\\frac{\\partial\\,B_{31}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial {z}} + \\frac{\\partial\\,E_{3}}{\\partial x}\\right], \\left[0, -\\frac{\\partial\\,B_{12}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial y} - \\frac{\\partial\\,E_{2}}{\\partial x}, 0, \\frac{\\partial\\,B_{23}}{\\partial {t}} - \\frac{\\partial\\,E_{2}}{\\partial {z}} + \\frac{\\partial\\,E_{3}}{\\partial y}\\right], \\left[0, \\frac{\\partial\\,B_{31}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial {z}} - \\frac{\\partial\\,E_{3}}{\\partial x}, -\\frac{\\partial\\,B_{23}}{\\partial {t}} + \\frac{\\partial\\,E_{2}}{\\partial {z}} - \\frac{\\partial\\,E_{3}}{\\partial y}, 0\\right]\\right], \\left[\\left[0, 0, -\\frac{\\partial\\,B_{12}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial y} - \\frac{\\partial\\,E_{2}}{\\partial x}, \\frac{\\partial\\,B_{31}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial {z}} - \\frac{\\partial\\,E_{3}}{\\partial x}\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{\\partial\\,B_{12}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial y} + \\frac{\\partial\\,E_{2}}{\\partial x}, 0, 0, \\frac{\\partial\\,B_{12}}{\\partial {z}} + \\frac{\\partial\\,B_{23}}{\\partial x} + \\frac{\\partial\\,B_{31}}{\\partial y}\\right], \\left[-\\frac{\\partial\\,B_{31}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial {z}} + \\frac{\\partial\\,E_{3}}{\\partial x}, 0, -\\frac{\\partial\\,B_{12}}{\\partial {z}} - \\frac{\\partial\\,B_{23}}{\\partial x} - \\frac{\\partial\\,B_{31}}{\\partial y}, 0\\right]\\right], \\left[\\left[0, \\frac{\\partial\\,B_{12}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial y} + \\frac{\\partial\\,E_{2}}{\\partial x}, 0, -\\frac{\\partial\\,B_{23}}{\\partial {t}} + \\frac{\\partial\\,E_{2}}{\\partial {z}} - \\frac{\\partial\\,E_{3}}{\\partial y}\\right], \\left[-\\frac{\\partial\\,B_{12}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial y} - \\frac{\\partial\\,E_{2}}{\\partial x}, 0, 0, -\\frac{\\partial\\,B_{12}}{\\partial {z}} - \\frac{\\partial\\,B_{23}}{\\partial x} - \\frac{\\partial\\,B_{31}}{\\partial y}\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{\\partial\\,B_{23}}{\\partial {t}} - \\frac{\\partial\\,E_{2}}{\\partial {z}} + \\frac{\\partial\\,E_{3}}{\\partial y}, \\frac{\\partial\\,B_{12}}{\\partial {z}} + \\frac{\\partial\\,B_{23}}{\\partial x} + \\frac{\\partial\\,B_{31}}{\\partial y}, 0, 0\\right]\\right], \\left[\\left[0, -\\frac{\\partial\\,B_{31}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial {z}} + \\frac{\\partial\\,E_{3}}{\\partial x}, \\frac{\\partial\\,B_{23}}{\\partial {t}} - \\frac{\\partial\\,E_{2}}{\\partial {z}} + \\frac{\\partial\\,E_{3}}{\\partial y}, 0\\right], \\left[\\frac{\\partial\\,B_{31}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial {z}} - \\frac{\\partial\\,E_{3}}{\\partial x}, 0, \\frac{\\partial\\,B_{12}}{\\partial {z}} + \\frac{\\partial\\,B_{23}}{\\partial x} + \\frac{\\partial\\,B_{31}}{\\partial y}, 0\\right], \\left[-\\frac{\\partial\\,B_{23}}{\\partial {t}} + \\frac{\\partial\\,E_{2}}{\\partial {z}} - \\frac{\\partial\\,E_{3}}{\\partial y}, -\\frac{\\partial\\,B_{12}}{\\partial {z}} - \\frac{\\partial\\,B_{23}}{\\partial x} - \\frac{\\partial\\,B_{31}}{\\partial y}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right]\\right]</script></html>"
      ],
      "text/plain": [
       "[[[0, 0, 0, 0],\n",
       "  [0, 0, d(B_12)/dt - d(E1)/dy + d(E2)/dx, -d(B_31)/dt - d(E1)/dz + d(E3)/dx],\n",
       "  [0, -d(B_12)/dt + d(E1)/dy - d(E2)/dx, 0, d(B_23)/dt - d(E2)/dz + d(E3)/dy],\n",
       "  [0, d(B_31)/dt + d(E1)/dz - d(E3)/dx, -d(B_23)/dt + d(E2)/dz - d(E3)/dy, 0]],\n",
       " [[0, 0, -d(B_12)/dt + d(E1)/dy - d(E2)/dx, d(B_31)/dt + d(E1)/dz - d(E3)/dx],\n",
       "  [0, 0, 0, 0],\n",
       "  [d(B_12)/dt - d(E1)/dy + d(E2)/dx,\n",
       "   0,\n",
       "   0,\n",
       "   d(B_12)/dz + d(B_23)/dx + d(B_31)/dy],\n",
       "  [-d(B_31)/dt - d(E1)/dz + d(E3)/dx,\n",
       "   0,\n",
       "   -d(B_12)/dz - d(B_23)/dx - d(B_31)/dy,\n",
       "   0]],\n",
       " [[0, d(B_12)/dt - d(E1)/dy + d(E2)/dx, 0, -d(B_23)/dt + d(E2)/dz - d(E3)/dy],\n",
       "  [-d(B_12)/dt + d(E1)/dy - d(E2)/dx,\n",
       "   0,\n",
       "   0,\n",
       "   -d(B_12)/dz - d(B_23)/dx - d(B_31)/dy],\n",
       "  [0, 0, 0, 0],\n",
       "  [d(B_23)/dt - d(E2)/dz + d(E3)/dy,\n",
       "   d(B_12)/dz + d(B_23)/dx + d(B_31)/dy,\n",
       "   0,\n",
       "   0]],\n",
       " [[0, -d(B_31)/dt - d(E1)/dz + d(E3)/dx, d(B_23)/dt - d(E2)/dz + d(E3)/dy, 0],\n",
       "  [d(B_31)/dt + d(E1)/dz - d(E3)/dx,\n",
       "   0,\n",
       "   d(B_12)/dz + d(B_23)/dx + d(B_31)/dy,\n",
       "   0],\n",
       "  [-d(B_23)/dt + d(E2)/dz - d(E3)/dy,\n",
       "   -d(B_12)/dz - d(B_23)/dx - d(B_31)/dy,\n",
       "   0,\n",
       "   0],\n",
       "  [0, 0, 0, 0]]]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EM_F.exterior_der()[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And so  \n",
    "$ dF = \\left[\\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, \\frac{\\partial\\,B_{12}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial y} + \\frac{\\partial\\,E_{2}}{\\partial x}, -\\frac{\\partial\\,B_{31}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial z} + \\frac{\\partial\\,E_{3}}{\\partial x}\\right], \\left[0, -\\frac{\\partial\\,B_{12}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial y} - \\frac{\\partial\\,E_{2}}{\\partial x}, 0, \\frac{\\partial\\,B_{23}}{\\partial {t}} - \\frac{\\partial\\,E_{2}}{\\partial z} + \\frac{\\partial\\,E_{3}}{\\partial y}\\right], \\left[0, \\frac{\\partial\\,B_{31}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial z} - \\frac{\\partial\\,E_{3}}{\\partial x}, -\\frac{\\partial\\,B_{23}}{\\partial {t}} + \\frac{\\partial\\,E_{2}}{\\partial z} - \\frac{\\partial\\,E_{3}}{\\partial y}, 0\\right]\\right], \\left[\\left[0, 0, -\\frac{\\partial\\,B_{12}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial y} - \\frac{\\partial\\,E_{2}}{\\partial x}, \\frac{\\partial\\,B_{31}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial z} - \\frac{\\partial\\,E_{3}}{\\partial x}\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{\\partial\\,B_{12}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial y} + \\frac{\\partial\\,E_{2}}{\\partial x}, 0, 0, \\frac{\\partial\\,B_{12}}{\\partial z} + \\frac{\\partial\\,B_{23}}{\\partial x} + \\frac{\\partial\\,B_{31}}{\\partial y}\\right], \\left[-\\frac{\\partial\\,B_{31}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial z} + \\frac{\\partial\\,E_{3}}{\\partial x}, 0, -\\frac{\\partial\\,B_{12}}{\\partial z} - \\frac{\\partial\\,B_{23}}{\\partial x} - \\frac{\\partial\\,B_{31}}{\\partial y}, 0\\right]\\right], \\left[\\left[0, \\frac{\\partial\\,B_{12}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial y} + \\frac{\\partial\\,E_{2}}{\\partial x}, 0, -\\frac{\\partial\\,B_{23}}{\\partial {t}} + \\frac{\\partial\\,E_{2}}{\\partial z} - \\frac{\\partial\\,E_{3}}{\\partial y}\\right], \\left[-\\frac{\\partial\\,B_{12}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial y} - \\frac{\\partial\\,E_{2}}{\\partial x}, 0, 0, -\\frac{\\partial\\,B_{12}}{\\partial z} - \\frac{\\partial\\,B_{23}}{\\partial x} - \\frac{\\partial\\,B_{31}}{\\partial y}\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{\\partial\\,B_{23}}{\\partial {t}} - \\frac{\\partial\\,E_{2}}{\\partial z} + \\frac{\\partial\\,E_{3}}{\\partial y}, \\frac{\\partial\\,B_{12}}{\\partial z} + \\frac{\\partial\\,B_{23}}{\\partial x} + \\frac{\\partial\\,B_{31}}{\\partial y}, 0, 0\\right]\\right], \\left[\\left[0, -\\frac{\\partial\\,B_{31}}{\\partial {t}} - \\frac{\\partial\\,E_{1}}{\\partial z} + \\frac{\\partial\\,E_{3}}{\\partial x}, \\frac{\\partial\\,B_{23}}{\\partial {t}} - \\frac{\\partial\\,E_{2}}{\\partial z} + \\frac{\\partial\\,E_{3}}{\\partial y}, 0\\right], \\left[\\frac{\\partial\\,B_{31}}{\\partial {t}} + \\frac{\\partial\\,E_{1}}{\\partial z} - \\frac{\\partial\\,E_{3}}{\\partial x}, 0, \\frac{\\partial\\,B_{12}}{\\partial z} + \\frac{\\partial\\,B_{23}}{\\partial x} + \\frac{\\partial\\,B_{31}}{\\partial y}, 0\\right], \\left[-\\frac{\\partial\\,B_{23}}{\\partial {t}} + \\frac{\\partial\\,E_{2}}{\\partial z} - \\frac{\\partial\\,E_{3}}{\\partial y}, -\\frac{\\partial\\,B_{12}}{\\partial z} - \\frac{\\partial\\,B_{23}}{\\partial x} - \\frac{\\partial\\,B_{31}}{\\partial y}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right]\\right] = 0  $\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "0 & B_{23}\\left({t}, x, y, {z}\\right) & B_{31}\\left({t}, x, y, {z}\\right) & B_{12}\\left({t}, x, y, {z}\\right) \\\\\n",
       "-B_{23}\\left({t}, x, y, {z}\\right) & 0 & E_{3}\\left({t}, x, y, {z}\\right) & -E_{2}\\left({t}, x, y, {z}\\right) \\\\\n",
       "-B_{31}\\left({t}, x, y, {z}\\right) & -E_{3}\\left({t}, x, y, {z}\\right) & 0 & E_{1}\\left({t}, x, y, {z}\\right) \\\\\n",
       "-B_{12}\\left({t}, x, y, {z}\\right) & E_{2}\\left({t}, x, y, {z}\\right) & -E_{1}\\left({t}, x, y, {z}\\right) & 0\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[                0  B_23(t, x, y, z)  B_31(t, x, y, z)  B_12(t, x, y, z)]\n",
       "[-B_23(t, x, y, z)                 0    E3(t, x, y, z)   -E2(t, x, y, z)]\n",
       "[-B_31(t, x, y, z)   -E3(t, x, y, z)                 0    E1(t, x, y, z)]\n",
       "[-B_12(t, x, y, z)    E2(t, x, y, z)   -E1(t, x, y, z)                 0]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EM_F.hodge_dual(g)[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a reminder, the Minkowski metric is a *Lorentzian metric* (not a Riemannian metric, which, in `sagemanifolds`, translates into using the lorentzian_metric module/function, not the riemannian_metric module/function) and is given by $g$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
       "-1 & 0 & 0 & 0 \\\\\n",
       "0 & 1 & 0 & 0 \\\\\n",
       "0 & 0 & 1 & 0 \\\\\n",
       "0 & 0 & 0 & 1\n",
       "\\end{array}\\right)</script></html>"
      ],
      "text/plain": [
       "[-1  0  0  0]\n",
       "[ 0  1  0  0]\n",
       "[ 0  0  1  0]\n",
       "[ 0  0  0  1]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$d*F$ (for $d*F = 4 \\pi * J$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, \\frac{\\partial\\,B_{23}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial x} + \\frac{\\partial\\,E_{3}}{\\partial {t}}, -\\frac{\\partial\\,B_{12}}{\\partial x} + \\frac{\\partial\\,B_{23}}{\\partial {z}} - \\frac{\\partial\\,E_{2}}{\\partial {t}}\\right], \\left[0, -\\frac{\\partial\\,B_{23}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial x} - \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, -\\frac{\\partial\\,B_{12}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial {z}} + \\frac{\\partial\\,E_{1}}{\\partial {t}}\\right], \\left[0, \\frac{\\partial\\,B_{12}}{\\partial x} - \\frac{\\partial\\,B_{23}}{\\partial {z}} + \\frac{\\partial\\,E_{2}}{\\partial {t}}, \\frac{\\partial\\,B_{12}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial {z}} - \\frac{\\partial\\,E_{1}}{\\partial {t}}, 0\\right]\\right], \\left[\\left[0, 0, -\\frac{\\partial\\,B_{23}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial x} - \\frac{\\partial\\,E_{3}}{\\partial {t}}, \\frac{\\partial\\,B_{12}}{\\partial x} - \\frac{\\partial\\,B_{23}}{\\partial {z}} + \\frac{\\partial\\,E_{2}}{\\partial {t}}\\right], \\left[0, 0, 0, 0\\right], \\left[\\frac{\\partial\\,B_{23}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial x} + \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, 0, \\frac{\\partial\\,E_{1}}{\\partial x} + \\frac{\\partial\\,E_{2}}{\\partial y} + \\frac{\\partial\\,E_{3}}{\\partial {z}}\\right], \\left[-\\frac{\\partial\\,B_{12}}{\\partial x} + \\frac{\\partial\\,B_{23}}{\\partial {z}} - \\frac{\\partial\\,E_{2}}{\\partial {t}}, 0, -\\frac{\\partial\\,E_{1}}{\\partial x} - \\frac{\\partial\\,E_{2}}{\\partial y} - \\frac{\\partial\\,E_{3}}{\\partial {z}}, 0\\right]\\right], \\left[\\left[0, \\frac{\\partial\\,B_{23}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial x} + \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, \\frac{\\partial\\,B_{12}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial {z}} - \\frac{\\partial\\,E_{1}}{\\partial {t}}\\right], \\left[-\\frac{\\partial\\,B_{23}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial x} - \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, 0, -\\frac{\\partial\\,E_{1}}{\\partial x} - \\frac{\\partial\\,E_{2}}{\\partial y} - \\frac{\\partial\\,E_{3}}{\\partial {z}}\\right], \\left[0, 0, 0, 0\\right], \\left[-\\frac{\\partial\\,B_{12}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial {z}} + \\frac{\\partial\\,E_{1}}{\\partial {t}}, \\frac{\\partial\\,E_{1}}{\\partial x} + \\frac{\\partial\\,E_{2}}{\\partial y} + \\frac{\\partial\\,E_{3}}{\\partial {z}}, 0, 0\\right]\\right], \\left[\\left[0, -\\frac{\\partial\\,B_{12}}{\\partial x} + \\frac{\\partial\\,B_{23}}{\\partial {z}} - \\frac{\\partial\\,E_{2}}{\\partial {t}}, -\\frac{\\partial\\,B_{12}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial {z}} + \\frac{\\partial\\,E_{1}}{\\partial {t}}, 0\\right], \\left[\\frac{\\partial\\,B_{12}}{\\partial x} - \\frac{\\partial\\,B_{23}}{\\partial {z}} + \\frac{\\partial\\,E_{2}}{\\partial {t}}, 0, \\frac{\\partial\\,E_{1}}{\\partial x} + \\frac{\\partial\\,E_{2}}{\\partial y} + \\frac{\\partial\\,E_{3}}{\\partial {z}}, 0\\right], \\left[\\frac{\\partial\\,B_{12}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial {z}} - \\frac{\\partial\\,E_{1}}{\\partial {t}}, -\\frac{\\partial\\,E_{1}}{\\partial x} - \\frac{\\partial\\,E_{2}}{\\partial y} - \\frac{\\partial\\,E_{3}}{\\partial {z}}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right]\\right]</script></html>"
      ],
      "text/plain": [
       "[[[0, 0, 0, 0],\n",
       "  [0,\n",
       "   0,\n",
       "   d(B_23)/dy - d(B_31)/dx + d(E3)/dt,\n",
       "   -d(B_12)/dx + d(B_23)/dz - d(E2)/dt],\n",
       "  [0,\n",
       "   -d(B_23)/dy + d(B_31)/dx - d(E3)/dt,\n",
       "   0,\n",
       "   -d(B_12)/dy + d(B_31)/dz + d(E1)/dt],\n",
       "  [0,\n",
       "   d(B_12)/dx - d(B_23)/dz + d(E2)/dt,\n",
       "   d(B_12)/dy - d(B_31)/dz - d(E1)/dt,\n",
       "   0]],\n",
       " [[0,\n",
       "   0,\n",
       "   -d(B_23)/dy + d(B_31)/dx - d(E3)/dt,\n",
       "   d(B_12)/dx - d(B_23)/dz + d(E2)/dt],\n",
       "  [0, 0, 0, 0],\n",
       "  [d(B_23)/dy - d(B_31)/dx + d(E3)/dt, 0, 0, d(E1)/dx + d(E2)/dy + d(E3)/dz],\n",
       "  [-d(B_12)/dx + d(B_23)/dz - d(E2)/dt,\n",
       "   0,\n",
       "   -d(E1)/dx - d(E2)/dy - d(E3)/dz,\n",
       "   0]],\n",
       " [[0,\n",
       "   d(B_23)/dy - d(B_31)/dx + d(E3)/dt,\n",
       "   0,\n",
       "   d(B_12)/dy - d(B_31)/dz - d(E1)/dt],\n",
       "  [-d(B_23)/dy + d(B_31)/dx - d(E3)/dt, 0, 0, -d(E1)/dx - d(E2)/dy - d(E3)/dz],\n",
       "  [0, 0, 0, 0],\n",
       "  [-d(B_12)/dy + d(B_31)/dz + d(E1)/dt, d(E1)/dx + d(E2)/dy + d(E3)/dz, 0, 0]],\n",
       " [[0,\n",
       "   -d(B_12)/dx + d(B_23)/dz - d(E2)/dt,\n",
       "   -d(B_12)/dy + d(B_31)/dz + d(E1)/dt,\n",
       "   0],\n",
       "  [d(B_12)/dx - d(B_23)/dz + d(E2)/dt, 0, d(E1)/dx + d(E2)/dy + d(E3)/dz, 0],\n",
       "  [d(B_12)/dy - d(B_31)/dz - d(E1)/dt, -d(E1)/dx - d(E2)/dy - d(E3)/dz, 0, 0],\n",
       "  [0, 0, 0, 0]]]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EM_F.hodge_dual(g).exterior_der()[:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a reminder, if you wanted the LaTeX code, use the `latex` function in Sage Math like such:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "latex( EM_F.hodge_dual(g).exterior_der()[:] ); # delete the semi-colon ; and you can get the LaTeX code-I suppress it"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And so  \n",
    "$d*F = \\left[\\left[\\left[0, 0, 0, 0\\right], \\left[0, 0, i \\, \\frac{\\partial\\,B_{23}}{\\partial y} - i \\, \\frac{\\partial\\,B_{31}}{\\partial x} + i \\, \\frac{\\partial\\,E_{3}}{\\partial {t}}, -i \\, \\frac{\\partial\\,B_{12}}{\\partial x} + i \\, \\frac{\\partial\\,B_{23}}{\\partial z} - i \\, \\frac{\\partial\\,E_{2}}{\\partial {t}}\\right], \\left[0, -i \\, \\frac{\\partial\\,B_{23}}{\\partial y} + i \\, \\frac{\\partial\\,B_{31}}{\\partial x} - i \\, \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, -i \\, \\frac{\\partial\\,B_{12}}{\\partial y} + i \\, \\frac{\\partial\\,B_{31}}{\\partial z} + i \\, \\frac{\\partial\\,E_{1}}{\\partial {t}}\\right], \\left[0, i \\, \\frac{\\partial\\,B_{12}}{\\partial x} - i \\, \\frac{\\partial\\,B_{23}}{\\partial z} + i \\, \\frac{\\partial\\,E_{2}}{\\partial {t}}, i \\, \\frac{\\partial\\,B_{12}}{\\partial y} - i \\, \\frac{\\partial\\,B_{31}}{\\partial z} - i \\, \\frac{\\partial\\,E_{1}}{\\partial {t}}, 0\\right]\\right], \\left[\\left[0, 0, -i \\, \\frac{\\partial\\,B_{23}}{\\partial y} + i \\, \\frac{\\partial\\,B_{31}}{\\partial x} - i \\, \\frac{\\partial\\,E_{3}}{\\partial {t}}, i \\, \\frac{\\partial\\,B_{12}}{\\partial x} - i \\, \\frac{\\partial\\,B_{23}}{\\partial z} + i \\, \\frac{\\partial\\,E_{2}}{\\partial {t}}\\right], \\left[0, 0, 0, 0\\right], \\left[i \\, \\frac{\\partial\\,B_{23}}{\\partial y} - i \\, \\frac{\\partial\\,B_{31}}{\\partial x} + i \\, \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, 0, i \\, \\frac{\\partial\\,E_{1}}{\\partial x} + i \\, \\frac{\\partial\\,E_{2}}{\\partial y} + i \\, \\frac{\\partial\\,E_{3}}{\\partial z}\\right], \\left[-i \\, \\frac{\\partial\\,B_{12}}{\\partial x} + i \\, \\frac{\\partial\\,B_{23}}{\\partial z} - i \\, \\frac{\\partial\\,E_{2}}{\\partial {t}}, 0, -i \\, \\frac{\\partial\\,E_{1}}{\\partial x} - i \\, \\frac{\\partial\\,E_{2}}{\\partial y} - i \\, \\frac{\\partial\\,E_{3}}{\\partial z}, 0\\right]\\right], \\left[\\left[0, i \\, \\frac{\\partial\\,B_{23}}{\\partial y} - i \\, \\frac{\\partial\\,B_{31}}{\\partial x} + i \\, \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, i \\, \\frac{\\partial\\,B_{12}}{\\partial y} - i \\, \\frac{\\partial\\,B_{31}}{\\partial z} - i \\, \\frac{\\partial\\,E_{1}}{\\partial {t}}\\right], \\left[-i \\, \\frac{\\partial\\,B_{23}}{\\partial y} + i \\, \\frac{\\partial\\,B_{31}}{\\partial x} - i \\, \\frac{\\partial\\,E_{3}}{\\partial {t}}, 0, 0, -i \\, \\frac{\\partial\\,E_{1}}{\\partial x} - i \\, \\frac{\\partial\\,E_{2}}{\\partial y} - i \\, \\frac{\\partial\\,E_{3}}{\\partial z}\\right], \\left[0, 0, 0, 0\\right], \\left[-i \\, \\frac{\\partial\\,B_{12}}{\\partial y} + i \\, \\frac{\\partial\\,B_{31}}{\\partial z} + i \\, \\frac{\\partial\\,E_{1}}{\\partial {t}}, i \\, \\frac{\\partial\\,E_{1}}{\\partial x} + i \\, \\frac{\\partial\\,E_{2}}{\\partial y} + i \\, \\frac{\\partial\\,E_{3}}{\\partial z}, 0, 0\\right]\\right], \\left[\\left[0, -i \\, \\frac{\\partial\\,B_{12}}{\\partial x} + i \\, \\frac{\\partial\\,B_{23}}{\\partial z} - i \\, \\frac{\\partial\\,E_{2}}{\\partial {t}}, -i \\, \\frac{\\partial\\,B_{12}}{\\partial y} + i \\, \\frac{\\partial\\,B_{31}}{\\partial z} + i \\, \\frac{\\partial\\,E_{1}}{\\partial {t}}, 0\\right], \\left[i \\, \\frac{\\partial\\,B_{12}}{\\partial x} - i \\, \\frac{\\partial\\,B_{23}}{\\partial z} + i \\, \\frac{\\partial\\,E_{2}}{\\partial {t}}, 0, i \\, \\frac{\\partial\\,E_{1}}{\\partial x} + i \\, \\frac{\\partial\\,E_{2}}{\\partial y} + i \\, \\frac{\\partial\\,E_{3}}{\\partial z}, 0\\right], \\left[i \\, \\frac{\\partial\\,B_{12}}{\\partial y} - i \\, \\frac{\\partial\\,B_{31}}{\\partial z} - i \\, \\frac{\\partial\\,E_{1}}{\\partial {t}}, -i \\, \\frac{\\partial\\,E_{1}}{\\partial x} - i \\, \\frac{\\partial\\,E_{2}}{\\partial y} - i \\, \\frac{\\partial\\,E_{3}}{\\partial z}, 0, 0\\right], \\left[0, 0, 0, 0\\right]\\right]\\right]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-\\frac{\\partial\\,E_{1}}{\\partial x} - \\frac{\\partial\\,E_{2}}{\\partial y} - \\frac{\\partial\\,E_{3}}{\\partial {z}}, \\frac{\\partial\\,B_{12}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial {z}} - \\frac{\\partial\\,E_{1}}{\\partial {t}}, -\\frac{\\partial\\,B_{12}}{\\partial x} + \\frac{\\partial\\,B_{23}}{\\partial {z}} - \\frac{\\partial\\,E_{2}}{\\partial {t}}, -\\frac{\\partial\\,B_{23}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial x} - \\frac{\\partial\\,E_{3}}{\\partial {t}}\\right]</script></html>"
      ],
      "text/plain": [
       "[-d(E1)/dx - d(E2)/dy - d(E3)/dz,\n",
       " d(B_12)/dy - d(B_31)/dz - d(E1)/dt,\n",
       " -d(B_12)/dx + d(B_23)/dz - d(E2)/dt,\n",
       " -d(B_23)/dy + d(B_31)/dx - d(E3)/dt]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "EM_F.hodge_dual(g).exterior_der().hodge_dual(g)[:] "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\\left[\\frac{\\partial\\,E_{1}}{\\partial x} + \\frac{\\partial\\,E_{2}}{\\partial y} + \\frac{\\partial\\,E_{3}}{\\partial z}, -\\frac{\\partial\\,B_{12}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial z} + \\frac{\\partial\\,E_{1}}{\\partial {t}}, \\frac{\\partial\\,B_{12}}{\\partial x} - \\frac{\\partial\\,B_{23}}{\\partial z} + \\frac{\\partial\\,E_{2}}{\\partial {t}}, \\frac{\\partial\\,B_{23}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial x} + \\frac{\\partial\\,E_{3}}{\\partial {t}}\\right]"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(EM_F.hodge_dual(g).exterior_der().hodge_dual(g)[:]); \n",
    "# delete the semi-colon ; and you can get the LaTeX code-I suppress it"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And so  \n",
    "$*d*F = \\left[\\frac{\\partial\\,E_{1}}{\\partial x} + \\frac{\\partial\\,E_{2}}{\\partial y} + \\frac{\\partial\\,E_{3}}{\\partial z}, -\\frac{\\partial\\,B_{12}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial z} + \\frac{\\partial\\,E_{1}}{\\partial {t}}, \\frac{\\partial\\,B_{12}}{\\partial x} - \\frac{\\partial\\,B_{23}}{\\partial z} + \\frac{\\partial\\,E_{2}}{\\partial {t}}, \\frac{\\partial\\,B_{23}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial x} + \\frac{\\partial\\,E_{3}}{\\partial {t}}\\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Current 1-form, Current conservation, and the other side (Right-Hand Side (RHS)) of $d*F$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def make_J(ch):\n",
    "    \"\"\"\n",
    "    make_J = make_J(ch)\n",
    "    make_J creates a time-INDEPENDENT current as a 1-form\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    Jcomplst = []\n",
    "    for i in range(1,4):\n",
    "        Jarglst = ['j'+str(i),] + list(ch[1:])\n",
    "        Jcomplst.append( function(Jarglst[0])(*Jarglst[1:]) )\n",
    "    Jcomplst = [-function('rho')(*list(ch[1:])),] +Jcomplst\n",
    "    J = ch.domain().diff_form(1)\n",
    "    J[ch.frame(),:,ch] = Jcomplst\n",
    "    return J\n",
    "\n",
    "\n",
    "def make_Jt(ch):\n",
    "    \"\"\"\n",
    "    make_Jt = make_Jt(ch)\n",
    "    make_Jt creates a time-DEPENDENT current as a 1-form\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    Jcomplst = []\n",
    "    for i in range(1,4):\n",
    "        Jarglst = ['j'+str(i),] + list(ch[:])\n",
    "        Jcomplst.append( function(Jarglst[0])(*Jarglst[1:]) )\n",
    "    Jcomplst = [-function('rho')(*list(ch[:])),]+Jcomplst\n",
    "    J = ch.domain().diff_form(1)\n",
    "    J[ch.frame(),:,ch] = Jcomplst\n",
    "    return J\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-rho(t, x, y, z) dt + j1(t, x, y, z) dx + j2(t, x, y, z) dy + j3(t, x, y, z) dz\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\rho\\left({t}, r, {\\phi}, {z}\\right) \\mathrm{d} {t} + j_{1}\\left({t}, r, {\\phi}, {z}\\right) \\mathrm{d} r + j_{2}\\left({t}, r, {\\phi}, {z}\\right) \\mathrm{d} {\\phi} + j_{3}\\left({t}, r, {\\phi}, {z}\\right) \\mathrm{d} {z}</script></html>"
      ],
      "text/plain": [
       "-rho(t, r, phi, z) dt + j1(t, r, phi, z) dr + j2(t, r, phi, z) dphi + j3(t, r, phi, z) dz"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print make_Jt(cart_ch).display() # these are examples of displaying the 4-current as 1-form in \n",
    "                                    # Cartesian and cylindrical coordinates\n",
    "make_Jt(cyl_ch).display(cyl_ch.frame(),cyl_ch)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\rho\\left({t}, x, y, {z}\\right) \\mathrm{d} t + j_{1}\\left({t}, x, y, {z}\\right) \\mathrm{d} x + j_{2}\\left({t}, x, y, {z}\\right) \\mathrm{d} y + j_{3}\\left({t}, x, y, {z}\\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "-rho(t, x, y, z) dt + j1(t, x, y, z) dx + j2(t, x, y, z) dy + j3(t, x, y, z) dz"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_Jt(cart_ch).hodge_dual(g).hodge_dual(g).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So here, I had successfully shown that $*d*F = **J$ or $*d*F = 4\\pi ** J$ (in cgs units), thus recovering Gauss's law and Ampere's law."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "latex( make_Jt(cart_ch).hodge_dual(g).hodge_dual(g)[:]);\n",
    "# delete the semi-colon ; and you can get the LaTeX code-I suppress it"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\boxed{ \\left[\\frac{\\partial\\,E_{1}}{\\partial x} + \\frac{\\partial\\,E_{2}}{\\partial y} + \\frac{\\partial\\,E_{3}}{\\partial z}, -\\frac{\\partial\\,B_{12}}{\\partial y} + \\frac{\\partial\\,B_{31}}{\\partial z} + \\frac{\\partial\\,E_{1}}{\\partial {t}}, \\frac{\\partial\\,B_{12}}{\\partial x} - \\frac{\\partial\\,B_{23}}{\\partial z} + \\frac{\\partial\\,E_{2}}{\\partial {t}}, \\frac{\\partial\\,B_{23}}{\\partial y} - \\frac{\\partial\\,B_{31}}{\\partial x} + \\frac{\\partial\\,E_{3}}{\\partial {t}}\\right] = \\left[\\rho\\left({t}, x, y, z\\right), -j_{1}\\left({t}, x, y, z\\right), -j_{2}\\left({t}, x, y, z\\right), -j_{3}\\left({t}, x, y, z\\right)\\right] }$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Current conservation is easily calculated, $*d*J=0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} & M & \\longrightarrow & \\mathbb{R} \\\\ & \\left({t}, x, y, {z}\\right) & \\longmapsto & -\\frac{\\partial\\,j_{1}}{\\partial x} - \\frac{\\partial\\,j_{2}}{\\partial y} - \\frac{\\partial\\,j_{3}}{\\partial {z}} - \\frac{\\partial\\,\\rho}{\\partial {t}} \\end{array}</script></html>"
      ],
      "text/plain": [
       "M --> R\n",
       "(t, x, y, z) |--> -d(j1)/dx - d(j2)/dy - d(j3)/dz - d(rho)/dt"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_Jt(cart_ch).hodge_dual(g).exterior_der().hodge_dual(g).display(cart_ch)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lorentz Force"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def make_beta(ch):\n",
    "    \"\"\"\n",
    "    make_beta = make_beta(ch)\n",
    "    make_beta creates a time-INDEPENDENT velocity field\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    betacomplst = []\n",
    "    for i in range(1,4):\n",
    "        betaarglst = ['beta'+str(i),] + list(ch[1:])\n",
    "        betacomplst.append( function(betaarglst[0])(*betaarglst[1:]) )\n",
    "    betacomplst = [1,]+betacomplst\n",
    "    beta = ch.domain().vector_field()\n",
    "    beta[ch.frame(),:,ch] = betacomplst\n",
    "    return beta\n",
    "\n",
    "def make_betat(ch):\n",
    "    \"\"\"\n",
    "    make_betat = make_betat(ch)\n",
    "    make_betat creates a time-DEPENDENT velocity field\n",
    "    \n",
    "    INPUT/PARAMETER\n",
    "    ch = sagemanifold chart\n",
    "    \"\"\"\n",
    "    betacomplst = []\n",
    "    for i in range(1,4):\n",
    "        betaarglst = ['beta'+str(i),] + list(ch[:])\n",
    "        betacomplst.append( function(betaarglst[0])(*betaarglst[1:]) )\n",
    "    betacomplst = [1,]+betacomplst\n",
    "    beta = ch.domain().vector_field()\n",
    "    beta[ch.frame(),:,ch] = betacomplst\n",
    "    return beta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial t } + \\beta_{1}\\left(x, y, {z}\\right) \\frac{\\partial}{\\partial x } + \\beta_{2}\\left(x, y, {z}\\right) \\frac{\\partial}{\\partial y } + \\beta_{3}\\left(x, y, {z}\\right) \\frac{\\partial}{\\partial z }</script></html>"
      ],
      "text/plain": [
       "d/dt + beta1(x, y, z) d/dx + beta2(x, y, z) d/dy + beta3(x, y, z) d/dz"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_beta(cart_ch).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For interior products, you're going to have to dig into how sagemanifolds implements Tensor products, tensor contractions, and the use of index notation, as sagemanifolds doesn't have a \"stand-alone\" interior product function.  From my [EuclideanManifold.py](https://github.com/ernestyalumni/Propulsion/blob/master/Learn/EuclideanManifold.py) implementation in sagemanifolds, look at my curl function (`def curl`) as a template for implementing interior products.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -B_{12}\\left({t}, x, y, {z}\\right) \\beta_{2}\\left({t}, x, y, {z}\\right) + B_{31}\\left({t}, x, y, {z}\\right) \\beta_{3}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} x + \\left( B_{12}\\left({t}, x, y, {z}\\right) \\beta_{1}\\left({t}, x, y, {z}\\right) - B_{23}\\left({t}, x, y, {z}\\right) \\beta_{3}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} y + \\left( -B_{31}\\left({t}, x, y, {z}\\right) \\beta_{1}\\left({t}, x, y, {z}\\right) + B_{23}\\left({t}, x, y, {z}\\right) \\beta_{2}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "(-B_12(t, x, y, z)*beta2(t, x, y, z) + B_31(t, x, y, z)*beta3(t, x, y, z)) dx + (B_12(t, x, y, z)*beta1(t, x, y, z) - B_23(t, x, y, z)*beta3(t, x, y, z)) dy + (-B_31(t, x, y, z)*beta1(t, x, y, z) + B_23(t, x, y, z)*beta2(t, x, y, z)) dz"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "betaeg = make_betat(cart_ch)\n",
    "Beg = make_Bt(cart_ch)\n",
    "(betaeg['^i']*Beg['_ij']).display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we now have a prescription on how to implement both the **interior product** and the **curl** of 2 \"vectors\" -  \n",
    "if you want this:\n",
    "\n",
    "$-i_{\\mathbf{\\beta}} B$ which is the differential form version of $\\mathbf{\\beta} \\times B$ (curl), then do this in sagemanifolds:\n",
    "\n",
    "`-betaeg['^i']*Beg['_ij']`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "q = var('q',domain=\"real\") # define a single charge variable in Sage Math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "LorentzForce1form =  make_Et(cart_ch) - make_beta(cart_ch)['^i']*make_Bt(cart_ch)['_ij']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can multiply this differential 1-form with Sage Math variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "LorentzForce1form = q * LorentzForce1form"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The 1-form version of the Lorentz Force $F$ is given by the following, and keep in mind that we integrate 1-forms, we don't integrate vectors (because it goes back to how we transport vectors along a curve, and 1-forms either abscond, circumvent, this problem or is the most natural way to do integration on manifolds): "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( q B_{12}\\left({t}, x, y, {z}\\right) \\beta_{2}\\left(x, y, {z}\\right) - q B_{31}\\left({t}, x, y, {z}\\right) \\beta_{3}\\left(x, y, {z}\\right) + q E_{1}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} x + \\left( -q B_{12}\\left({t}, x, y, {z}\\right) \\beta_{1}\\left(x, y, {z}\\right) + q B_{23}\\left({t}, x, y, {z}\\right) \\beta_{3}\\left(x, y, {z}\\right) + q E_{2}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} y + \\left( q B_{31}\\left({t}, x, y, {z}\\right) \\beta_{1}\\left(x, y, {z}\\right) - q B_{23}\\left({t}, x, y, {z}\\right) \\beta_{2}\\left(x, y, {z}\\right) + q E_{3}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "(q*B_12(t, x, y, z)*beta2(x, y, z) - q*B_31(t, x, y, z)*beta3(x, y, z) + q*E1(t, x, y, z)) dx + (-q*B_12(t, x, y, z)*beta1(x, y, z) + q*B_23(t, x, y, z)*beta3(x, y, z) + q*E2(t, x, y, z)) dy + (q*B_31(t, x, y, z)*beta1(x, y, z) - q*B_23(t, x, y, z)*beta2(x, y, z) + q*E3(t, x, y, z)) dz"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LorentzForce1form.display() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And so  \n",
    "$f = \\left( q B_{12}\\left({t}, x, y, z\\right) \\beta_{2}\\left(x, y, z\\right) - q B_{31}\\left({t}, x, y, z\\right) \\beta_{3}\\left(x, y, z\\right) + q E_{1}\\left({t}, x, y, z\\right) \\right) \\mathrm{d} x + \\left( -q B_{12}\\left({t}, x, y, z\\right) \\beta_{1}\\left(x, y, z\\right) + q B_{23}\\left({t}, x, y, z\\right) \\beta_{3}\\left(x, y, z\\right) + q E_{2}\\left({t}, x, y, z\\right) \\right) \\mathrm{d} y + \\left( q B_{31}\\left({t}, x, y, z\\right) \\beta_{1}\\left(x, y, z\\right) - q B_{23}\\left({t}, x, y, z\\right) \\beta_{2}\\left(x, y, z\\right) + q E_{3}\\left({t}, x, y, z\\right) \\right) \\mathrm{d} z$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## More \"combinations\": $F \\wedge *F$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$F \\wedge * F$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "FwedgestarF = EM_F.wedge(EM_F.hodge_dual(g))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( B_{12}\\left({t}, x, y, {z}\\right)^{2} + B_{23}\\left({t}, x, y, {z}\\right)^{2} + B_{31}\\left({t}, x, y, {z}\\right)^{2} - E_{1}\\left({t}, x, y, {z}\\right)^{2} - E_{2}\\left({t}, x, y, {z}\\right)^{2} - E_{3}\\left({t}, x, y, {z}\\right)^{2} \\right) \\mathrm{d} t\\wedge \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "(B_12(t, x, y, z)^2 + B_23(t, x, y, z)^2 + B_31(t, x, y, z)^2 - E1(t, x, y, z)^2 - E2(t, x, y, z)^2 - E3(t, x, y, z)^2) dt/\\dx/\\dy/\\dz"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FwedgestarF.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$E\\cdot B$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( B_{23}\\left({t}, x, y, {z}\\right) E_{1}\\left({t}, x, y, {z}\\right) + B_{31}\\left({t}, x, y, {z}\\right) E_{2}\\left({t}, x, y, {z}\\right) + B_{12}\\left({t}, x, y, {z}\\right) E_{3}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "(B_23(t, x, y, z)*E1(t, x, y, z) + B_31(t, x, y, z)*E2(t, x, y, z) + B_12(t, x, y, z)*E3(t, x, y, z)) dx/\\dy/\\dz"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eteg = make_Et(cart_ch)\n",
    "Bteg = make_Bt(cart_ch)\n",
    "EdotB = Eteg.wedge( Bteg)\n",
    "EdotB.display()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Poynting Vector\n",
    "\n",
    "Try\n",
    "$E \\wedge *B$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( -B_{31}\\left({t}, x, y, {z}\\right) E_{1}\\left({t}, x, y, {z}\\right) + B_{23}\\left({t}, x, y, {z}\\right) E_{2}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} t\\wedge \\mathrm{d} x\\wedge \\mathrm{d} y + \\left( -B_{12}\\left({t}, x, y, {z}\\right) E_{1}\\left({t}, x, y, {z}\\right) + B_{23}\\left({t}, x, y, {z}\\right) E_{3}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} t\\wedge \\mathrm{d} x\\wedge \\mathrm{d} z + \\left( -B_{12}\\left({t}, x, y, {z}\\right) E_{2}\\left({t}, x, y, {z}\\right) + B_{31}\\left({t}, x, y, {z}\\right) E_{3}\\left({t}, x, y, {z}\\right) \\right) \\mathrm{d} t\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z</script></html>"
      ],
      "text/plain": [
       "(-B_31(t, x, y, z)*E1(t, x, y, z) + B_23(t, x, y, z)*E2(t, x, y, z)) dt/\\dx/\\dy + (-B_12(t, x, y, z)*E1(t, x, y, z) + B_23(t, x, y, z)*E3(t, x, y, z)) dt/\\dx/\\dz + (-B_12(t, x, y, z)*E2(t, x, y, z) + B_31(t, x, y, z)*E3(t, x, y, z)) dt/\\dy/\\dz"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(Eteg.wedge( Bteg.hodge_dual(g))).display()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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