examples/ipython/second_derivative.ipynb

{
 "cells": [
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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation*} phi = \\phi ^{t}  \\boldsymbol{e}_{t} + \\phi ^{x}  \\boldsymbol{e}_{x} + \\phi ^{y}  \\boldsymbol{e}_{y} + \\phi ^{z}  \\boldsymbol{e}_{z} \\end{equation*}"
      ],
      "text/plain": [
       "\\phi ^{t}  \\boldsymbol{e}_{t} + \\phi ^{x}  \\boldsymbol{e}_{x} + \\phi ^{y}  \\boldsymbol{e}_{y} + \\phi ^{z}  \\boldsymbol{e}_{z}"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from galgebra.ga import Ga\n",
    "from sympy import symbols\n",
    "from galgebra.printer import Format\n",
    "\n",
    "Format()\n",
    "coords = (et,ex,ey,ez) = symbols('t,x,y,z',real=True)\n",
    "base=Ga('e*t|x|y|z',g=[1,-1,-1,-1],coords=symbols('t,x,y,z',real=True),wedge=False)\n",
    "\n",
    "potential=base.mv('phi','vector',f=True)\n",
    "potential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation*} \\left ( \\partial_{t} \\phi ^{t}  + \\partial_{x} \\phi ^{x}  + \\partial_{y} \\phi ^{y}  + \\partial_{z} \\phi ^{z} \\right )  + \\left ( \\partial_{x} \\phi ^{t}  + \\partial_{t} \\phi ^{x} \\right ) \\boldsymbol{e}_{tx} + \\left ( \\partial_{y} \\phi ^{t}  + \\partial_{t} \\phi ^{y} \\right ) \\boldsymbol{e}_{ty} + \\left ( \\partial_{z} \\phi ^{t}  + \\partial_{t} \\phi ^{z} \\right ) \\boldsymbol{e}_{tz} + \\left ( \\partial_{y} \\phi ^{x}  - \\partial_{x} \\phi ^{y} \\right ) \\boldsymbol{e}_{xy} + \\left ( \\partial_{z} \\phi ^{x}  - \\partial_{x} \\phi ^{z} \\right ) \\boldsymbol{e}_{xz} + \\left ( \\partial_{z} \\phi ^{y}  - \\partial_{y} \\phi ^{z} \\right ) \\boldsymbol{e}_{yz} \\end{equation*}"
      ],
      "text/plain": [
       "\\left ( \\partial_{t} \\phi ^{t}  + \\partial_{x} \\phi ^{x}  + \\partial_{y} \\phi ^{y}  + \\partial_{z} \\phi ^{z} \\right )  + \\left ( \\partial_{x} \\phi ^{t}  + \\partial_{t} \\phi ^{x} \\right ) \\boldsymbol{e}_{tx} + \\left ( \\partial_{y} \\phi ^{t}  + \\partial_{t} \\phi ^{y} \\right ) \\boldsymbol{e}_{ty} + \\left ( \\partial_{z} \\phi ^{t}  + \\partial_{t} \\phi ^{z} \\right ) \\boldsymbol{e}_{tz} + \\left ( \\partial_{y} \\phi ^{x}  - \\partial_{x} \\phi ^{y} \\right ) \\boldsymbol{e}_{xy} + \\left ( \\partial_{z} \\phi ^{x}  - \\partial_{x} \\phi ^{z} \\right ) \\boldsymbol{e}_{xz} + \\left ( \\partial_{z} \\phi ^{y}  - \\partial_{y} \\phi ^{z} \\right ) \\boldsymbol{e}_{yz}"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "field=base.grad*potential\n",
    "field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation*} \\left ( \\partial^{2}_{t} \\phi ^{t}  - \\partial^{2}_{x} \\phi ^{t}  - \\partial^{2}_{y} \\phi ^{t}  - \\partial^{2}_{z} \\phi ^{t} \\right ) \\boldsymbol{e}_{t} + \\left ( \\partial^{2}_{t} \\phi ^{x}  - \\partial^{2}_{x} \\phi ^{x}  - \\partial^{2}_{y} \\phi ^{x}  - \\partial^{2}_{z} \\phi ^{x} \\right ) \\boldsymbol{e}_{x} + \\left ( \\partial^{2}_{t} \\phi ^{y}  - \\partial^{2}_{x} \\phi ^{y}  - \\partial^{2}_{y} \\phi ^{y}  - \\partial^{2}_{z} \\phi ^{y} \\right ) \\boldsymbol{e}_{y} + \\left ( \\partial^{2}_{t} \\phi ^{z}  - \\partial^{2}_{x} \\phi ^{z}  - \\partial^{2}_{y} \\phi ^{z}  - \\partial^{2}_{z} \\phi ^{z} \\right ) \\boldsymbol{e}_{z} \\end{equation*}"
      ],
      "text/plain": [
       "\\left ( \\partial^{2}_{t} \\phi ^{t}  - \\partial^{2}_{x} \\phi ^{t}  - \\partial^{2}_{y} \\phi ^{t}  - \\partial^{2}_{z} \\phi ^{t} \\right ) \\boldsymbol{e}_{t} + \\left ( \\partial^{2}_{t} \\phi ^{x}  - \\partial^{2}_{x} \\phi ^{x}  - \\partial^{2}_{y} \\phi ^{x}  - \\partial^{2}_{z} \\phi ^{x} \\right ) \\boldsymbol{e}_{x} + \\left ( \\partial^{2}_{t} \\phi ^{y}  - \\partial^{2}_{x} \\phi ^{y}  - \\partial^{2}_{y} \\phi ^{y}  - \\partial^{2}_{z} \\phi ^{y} \\right ) \\boldsymbol{e}_{y} + \\left ( \\partial^{2}_{t} \\phi ^{z}  - \\partial^{2}_{x} \\phi ^{z}  - \\partial^{2}_{y} \\phi ^{z}  - \\partial^{2}_{z} \\phi ^{z} \\right ) \\boldsymbol{e}_{z}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grad_field = base.grad*field\n",
    "grad_field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation*} \\left ( \\partial_{x} \\phi ^{t}  + \\partial_{t} \\phi ^{x} \\right ) \\boldsymbol{e}_{tx} \\end{equation*}"
      ],
      "text/plain": [
       "\\left ( \\partial_{x} \\phi ^{t}  + \\partial_{t} \\phi ^{x} \\right ) \\boldsymbol{e}_{tx}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "part=field.proj([base.mv()[0]^base.mv()[1]])\n",
    "part"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation*} \\left ( - \\partial^{2}_{x} \\phi ^{t}  - \\partial_{t}\\partial_{x} \\phi ^{x} \\right ) \\boldsymbol{e}_{t} + \\left ( \\partial^{2}_{t} \\phi ^{x}  + \\partial_{t}\\partial_{x} \\phi ^{t} \\right ) \\boldsymbol{e}_{x} + \\left ( - \\partial_{x}\\partial_{y} \\phi ^{t}  - \\partial_{t}\\partial_{y} \\phi ^{x} \\right ) \\boldsymbol{e}_{txy} + \\left ( - \\partial_{x}\\partial_{z} \\phi ^{t}  - \\partial_{t}\\partial_{z} \\phi ^{x} \\right ) \\boldsymbol{e}_{txz} \\end{equation*}"
      ],
      "text/plain": [
       "\\left ( - \\partial^{2}_{x} \\phi ^{t}  - \\partial_{t}\\partial_{x} \\phi ^{x} \\right ) \\boldsymbol{e}_{t} + \\left ( \\partial^{2}_{t} \\phi ^{x}  + \\partial_{t}\\partial_{x} \\phi ^{t} \\right ) \\boldsymbol{e}_{x} + \\left ( - \\partial_{x}\\partial_{y} \\phi ^{t}  - \\partial_{t}\\partial_{y} \\phi ^{x} \\right ) \\boldsymbol{e}_{txy} + \\left ( - \\partial_{x}\\partial_{z} \\phi ^{t}  - \\partial_{t}\\partial_{z} \\phi ^{x} \\right ) \\boldsymbol{e}_{txz}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dpart = base.grad*part\n",
    "dpart"
   ]
  }
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