{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%cd -q ../Terminal"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = A + A__x*\u001b[0;34me_x\u001b[0m + A__y*\u001b[0;34me_y\u001b[0m + A__z*\u001b[0;34me_z\u001b[0m + A__xy*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + A__xz*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + A__yz*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m + A__xyz*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"A = A\n",
" + A__x*\u001b[0;34me_x\u001b[0m + A__y*\u001b[0;34me_y\u001b[0m + A__z*\u001b[0;34me_z\u001b[0m\n",
" + A__xy*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + A__xz*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + A__yz*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
" + A__xyz*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"A = A\n",
" + A__x*\u001b[0;34me_x\u001b[0m\n",
" + A__y*\u001b[0;34me_y\u001b[0m\n",
" + A__z*\u001b[0;34me_z\u001b[0m\n",
" + A__xy*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m\n",
" + A__xz*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
" + A__yz*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
" + A__xyz*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"g_{ij} =\n",
" Matrix([\n",
"[(\u001b[0;34me_x\u001b[0m.\u001b[0;34me_x\u001b[0m), (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m), (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_z\u001b[0m)],\n",
"[(\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m), (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_y\u001b[0m), (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_z\u001b[0m)],\n",
"[(\u001b[0;34me_x\u001b[0m.\u001b[0;34me_z\u001b[0m), (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_z\u001b[0m), (\u001b[0;34me_z\u001b[0m.\u001b[0;34me_z\u001b[0m)]])\n",
"X = X__x*\u001b[0;34me_x\u001b[0m + X__y*\u001b[0;34me_y\u001b[0m + X__z*\u001b[0;34me_z\u001b[0m\n",
"Y = Y__x*\u001b[0;34me_x\u001b[0m + Y__y*\u001b[0;34me_y\u001b[0m + Y__z*\u001b[0;34me_z\u001b[0m\n",
"X*Y = (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_x\u001b[0m)*X__x*Y__x + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__x*Y__y + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y*Y__x + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__x*Y__z + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__z*Y__x + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y*Y__y + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__y*Y__z + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__z*Y__y + (\u001b[0;34me_z\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__z*Y__z\n",
" + (X__x*Y__y - X__y*Y__x)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + (X__x*Y__z - X__z*Y__x)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + (X__y*Y__z - X__z*Y__y)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"X^Y = (X__x*Y__y - X__y*Y__x)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + (X__x*Y__z - X__z*Y__x)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + (X__y*Y__z - X__z*Y__y)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"X|Y = (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_x\u001b[0m)*X__x*Y__x + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__x*Y__y + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y*Y__x + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__x*Y__z + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__z*Y__x + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y*Y__y + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__y*Y__z + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__z*Y__y + (\u001b[0;34me_z\u001b[0m.\u001b[0;34me_z\u001b[0m)*X__z*Y__z\n",
"g_{ij} =\n",
" Matrix([\n",
"[(\u001b[0;34me_x\u001b[0m.\u001b[0;34me_x\u001b[0m), (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)],\n",
"[(\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m), (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_y\u001b[0m)]])\n",
"X = X__x*\u001b[0;34me_x\u001b[0m + X__y*\u001b[0;34me_y\u001b[0m\n",
"A = A + A__xy*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m\n",
"X|A = -A__xy*((\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__x + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y)*\u001b[0;34me_x\u001b[0m + A__xy*((\u001b[0;34me_x\u001b[0m.\u001b[0;34me_x\u001b[0m)*X__x + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y)*\u001b[0;34me_y\u001b[0m\n",
"XX = A__xy*((\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__x + (\u001b[0;34me_y\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y)*\u001b[0;34me_x\u001b[0m - A__xy*((\u001b[0;34me_x\u001b[0m.\u001b[0;34me_x\u001b[0m)*X__x + (\u001b[0;34me_x\u001b[0m.\u001b[0;34me_y\u001b[0m)*X__y)*\u001b[0;34me_y\u001b[0m\n",
"g_{ii} =\n",
" Matrix([\n",
"[1, 0],\n",
"[0, 1]])\n",
"X = X__x*\u001b[0;34me_x\u001b[0m + X__y*\u001b[0;34me_y\u001b[0m\n",
"A = A + A__xy*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m\n",
"X*A = (A*X__x - A__xy*X__y)*\u001b[0;34me_x\u001b[0m + (A*X__y + A__xy*X__x)*\u001b[0;34me_y\u001b[0m\n",
"X|A = -A__xy*X__y*\u001b[0;34me_x\u001b[0m + A__xy*X__x*\u001b[0;34me_y\u001b[0m\n",
"XA = A*X__x*\u001b[0;34me_x\u001b[0m + A*X__y*\u001b[0;34me_y\u001b[0m\n",
"A*X = (A*X__x + A__xy*X__y)*\u001b[0;34me_x\u001b[0m + (A*X__y - A__xy*X__x)*\u001b[0;34me_y\u001b[0m\n",
"A|X = A__xy*X__y*\u001b[0;34me_x\u001b[0m - A__xy*X__x*\u001b[0;34me_y\u001b[0m\n",
"AX = A__xy*X__y*\u001b[0;34me_x\u001b[0m - A__xy*X__x*\u001b[0;34me_y\u001b[0m\n",
"g_{ij} =\n",
" Matrix([\n",
"[(\u001b[0;34ma\u001b[0m.\u001b[0;34ma\u001b[0m), (\u001b[0;34ma\u001b[0m.\u001b[0;34mb\u001b[0m), (\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m), (\u001b[0;34ma\u001b[0m.\u001b[0;34md\u001b[0m), (\u001b[0;34ma\u001b[0m.\u001b[0;34me\u001b[0m)],\n",
"[(\u001b[0;34ma\u001b[0m.\u001b[0;34mb\u001b[0m), (\u001b[0;34mb\u001b[0m.\u001b[0;34mb\u001b[0m), (\u001b[0;34mb\u001b[0m.\u001b[0;34mc\u001b[0m), (\u001b[0;34mb\u001b[0m.\u001b[0;34md\u001b[0m), (\u001b[0;34mb\u001b[0m.\u001b[0;34me\u001b[0m)],\n",
"[(\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m), (\u001b[0;34mb\u001b[0m.\u001b[0;34mc\u001b[0m), (\u001b[0;34mc\u001b[0m.\u001b[0;34mc\u001b[0m), (\u001b[0;34mc\u001b[0m.\u001b[0;34md\u001b[0m), (\u001b[0;34mc\u001b[0m.\u001b[0;34me\u001b[0m)],\n",
"[(\u001b[0;34ma\u001b[0m.\u001b[0;34md\u001b[0m), (\u001b[0;34mb\u001b[0m.\u001b[0;34md\u001b[0m), (\u001b[0;34mc\u001b[0m.\u001b[0;34md\u001b[0m), (\u001b[0;34md\u001b[0m.\u001b[0;34md\u001b[0m), (\u001b[0;34md\u001b[0m.\u001b[0;34me\u001b[0m)],\n",
"[(\u001b[0;34ma\u001b[0m.\u001b[0;34me\u001b[0m), (\u001b[0;34mb\u001b[0m.\u001b[0;34me\u001b[0m), (\u001b[0;34mc\u001b[0m.\u001b[0;34me\u001b[0m), (\u001b[0;34md\u001b[0m.\u001b[0;34me\u001b[0m), (\u001b[0;34me\u001b[0m.\u001b[0;34me\u001b[0m)]])\n",
"a|(b*c) = -(\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*\u001b[0;34mb\u001b[0m + (\u001b[0;34ma\u001b[0m.\u001b[0;34mb\u001b[0m)*\u001b[0;34mc\u001b[0m\n",
"a|(b^c) = -(\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*\u001b[0;34mb\u001b[0m + (\u001b[0;34ma\u001b[0m.\u001b[0;34mb\u001b[0m)*\u001b[0;34mc\u001b[0m\n",
"a|(b^c^d) = (\u001b[0;34ma\u001b[0m.\u001b[0;34md\u001b[0m)*\u001b[0;34mb\u001b[0m^\u001b[0;34mc\u001b[0m - (\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*\u001b[0;34mb\u001b[0m^\u001b[0;34md\u001b[0m + (\u001b[0;34ma\u001b[0m.\u001b[0;34mb\u001b[0m)*\u001b[0;34mc\u001b[0m^\u001b[0;34md\u001b[0m\n",
"a|(b^c)+c|(a^b)+b|(c^a) = 0\n",
"a*(b^c)-b*(a^c)+c*(a^b) = 3*\u001b[0;34ma\u001b[0m^\u001b[0;34mb\u001b[0m^\u001b[0;34mc\u001b[0m\n",
"a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) = 4*\u001b[0;34ma\u001b[0m^\u001b[0;34mb\u001b[0m^\u001b[0;34mc\u001b[0m^\u001b[0;34md\u001b[0m\n",
"(a^b)|(c^d) = -(\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*(\u001b[0;34mb\u001b[0m.\u001b[0;34md\u001b[0m) + (\u001b[0;34ma\u001b[0m.\u001b[0;34md\u001b[0m)*(\u001b[0;34mb\u001b[0m.\u001b[0;34mc\u001b[0m)\n",
"((a^b)|c)|d = -(\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*(\u001b[0;34mb\u001b[0m.\u001b[0;34md\u001b[0m) + (\u001b[0;34ma\u001b[0m.\u001b[0;34md\u001b[0m)*(\u001b[0;34mb\u001b[0m.\u001b[0;34mc\u001b[0m)\n",
"(a^b)x(c^d) = -(\u001b[0;34mb\u001b[0m.\u001b[0;34md\u001b[0m)*\u001b[0;34ma\u001b[0m^\u001b[0;34mc\u001b[0m + (\u001b[0;34mb\u001b[0m.\u001b[0;34mc\u001b[0m)*\u001b[0;34ma\u001b[0m^\u001b[0;34md\u001b[0m + (\u001b[0;34ma\u001b[0m.\u001b[0;34md\u001b[0m)*\u001b[0;34mb\u001b[0m^\u001b[0;34mc\u001b[0m - (\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*\u001b[0;34mb\u001b[0m^\u001b[0;34md\u001b[0m\n",
"(a|(b^c))|(d^e) = (-(\u001b[0;34ma\u001b[0m.\u001b[0;34mb\u001b[0m)*(\u001b[0;34mc\u001b[0m.\u001b[0;34me\u001b[0m) + (\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*(\u001b[0;34mb\u001b[0m.\u001b[0;34me\u001b[0m))*\u001b[0;34md\u001b[0m + ((\u001b[0;34ma\u001b[0m.\u001b[0;34mb\u001b[0m)*(\u001b[0;34mc\u001b[0m.\u001b[0;34md\u001b[0m) - (\u001b[0;34ma\u001b[0m.\u001b[0;34mc\u001b[0m)*(\u001b[0;34mb\u001b[0m.\u001b[0;34md\u001b[0m))*\u001b[0;34me\u001b[0m\n",
"f = \u001b[0;31mf\u001b[0m\n",
"A = \u001b[0;31mA__x\u001b[0m*\u001b[0;34me_x\u001b[0m + \u001b[0;31mA__y\u001b[0m*\u001b[0;34me_y\u001b[0m + \u001b[0;31mA__z\u001b[0m*\u001b[0;34me_z\u001b[0m\n",
"B = \u001b[0;31mB__xy\u001b[0m*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + \u001b[0;31mB__xz\u001b[0m*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + \u001b[0;31mB__yz\u001b[0m*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"C = \u001b[0;31mC\u001b[0m + \u001b[0;31mC__x\u001b[0m*\u001b[0;34me_x\u001b[0m + \u001b[0;31mC__y\u001b[0m*\u001b[0;34me_y\u001b[0m + \u001b[0;31mC__z\u001b[0m*\u001b[0;34me_z\u001b[0m + \u001b[0;31mC__xy\u001b[0m*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + \u001b[0;31mC__xz\u001b[0m*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + \u001b[0;31mC__yz\u001b[0m*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m + \u001b[0;31mC__xyz\u001b[0m*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"grad*f = \u001b[0;36mD{x}\u001b[0;31mf\u001b[0m\u001b[0m*\u001b[0;34me_x\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mf\u001b[0m\u001b[0m*\u001b[0;34me_y\u001b[0m + \u001b[0;36mD{z}\u001b[0;31mf\u001b[0m\u001b[0m*\u001b[0;34me_z\u001b[0m\n",
"grad|A = \u001b[0;36mD{x}\u001b[0;31mA__x\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mA__y\u001b[0m\u001b[0m + \u001b[0;36mD{z}\u001b[0;31mA__z\u001b[0m\u001b[0m\n",
"grad*A = \u001b[0;36mD{x}\u001b[0;31mA__x\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mA__y\u001b[0m\u001b[0m + \u001b[0;36mD{z}\u001b[0;31mA__z\u001b[0m\u001b[0m + (-\u001b[0;36mD{y}\u001b[0;31mA__x\u001b[0m\u001b[0m + \u001b[0;36mD{x}\u001b[0;31mA__y\u001b[0m\u001b[0m)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + (-\u001b[0;36mD{z}\u001b[0;31mA__x\u001b[0m\u001b[0m + \u001b[0;36mD{x}\u001b[0;31mA__z\u001b[0m\u001b[0m)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + (-\u001b[0;36mD{z}\u001b[0;31mA__y\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mA__z\u001b[0m\u001b[0m)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"-I*(grad^A) = (-\u001b[0;36mD{z}\u001b[0;31mA__y\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mA__z\u001b[0m\u001b[0m)*\u001b[0;34me_x\u001b[0m + (\u001b[0;36mD{z}\u001b[0;31mA__x\u001b[0m\u001b[0m - \u001b[0;36mD{x}\u001b[0;31mA__z\u001b[0m\u001b[0m)*\u001b[0;34me_y\u001b[0m + (-\u001b[0;36mD{y}\u001b[0;31mA__x\u001b[0m\u001b[0m + \u001b[0;36mD{x}\u001b[0;31mA__y\u001b[0m\u001b[0m)*\u001b[0;34me_z\u001b[0m\n",
"grad*B = (-\u001b[0;36mD{y}\u001b[0;31mB__xy\u001b[0m\u001b[0m - \u001b[0;36mD{z}\u001b[0;31mB__xz\u001b[0m\u001b[0m)*\u001b[0;34me_x\u001b[0m + (\u001b[0;36mD{x}\u001b[0;31mB__xy\u001b[0m\u001b[0m - \u001b[0;36mD{z}\u001b[0;31mB__yz\u001b[0m\u001b[0m)*\u001b[0;34me_y\u001b[0m + (\u001b[0;36mD{x}\u001b[0;31mB__xz\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mB__yz\u001b[0m\u001b[0m)*\u001b[0;34me_z\u001b[0m + (\u001b[0;36mD{z}\u001b[0;31mB__xy\u001b[0m\u001b[0m - \u001b[0;36mD{y}\u001b[0;31mB__xz\u001b[0m\u001b[0m + \u001b[0;36mD{x}\u001b[0;31mB__yz\u001b[0m\u001b[0m)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"grad^B = (\u001b[0;36mD{z}\u001b[0;31mB__xy\u001b[0m\u001b[0m - \u001b[0;36mD{y}\u001b[0;31mB__xz\u001b[0m\u001b[0m + \u001b[0;36mD{x}\u001b[0;31mB__yz\u001b[0m\u001b[0m)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"grad|B = (-\u001b[0;36mD{y}\u001b[0;31mB__xy\u001b[0m\u001b[0m - \u001b[0;36mD{z}\u001b[0;31mB__xz\u001b[0m\u001b[0m)*\u001b[0;34me_x\u001b[0m + (\u001b[0;36mD{x}\u001b[0;31mB__xy\u001b[0m\u001b[0m - \u001b[0;36mD{z}\u001b[0;31mB__yz\u001b[0m\u001b[0m)*\u001b[0;34me_y\u001b[0m + (\u001b[0;36mD{x}\u001b[0;31mB__xz\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mB__yz\u001b[0m\u001b[0m)*\u001b[0;34me_z\u001b[0m\n",
"gradA = \u001b[0;36mD{x}\u001b[0;31mA__x\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mA__y\u001b[0m\u001b[0m + \u001b[0;36mD{z}\u001b[0;31mA__z\u001b[0m\u001b[0m\n",
"gradB = 0\n",
"gradC = \u001b[0;36mD{x}\u001b[0;31mC__x\u001b[0m\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mC__y\u001b[0m\u001b[0m + \u001b[0;36mD{z}\u001b[0;31mC__z\u001b[0m\u001b[0m + \u001b[0;36mD{x}\u001b[0;31mC\u001b[0m\u001b[0m*\u001b[0;34me_x\u001b[0m + \u001b[0;36mD{y}\u001b[0;31mC\u001b[0m\u001b[0m*\u001b[0;34me_y\u001b[0m + \u001b[0;36mD{z}\u001b[0;31mC\u001b[0m\u001b[0m*\u001b[0;34me_z\u001b[0m\n",
"f = \u001b[0;31mf\u001b[0m\n",
"A = \u001b[0;31mA__r\u001b[0m*\u001b[0;34me_r\u001b[0m + \u001b[0;31mA__theta\u001b[0m*\u001b[0;34me_theta\u001b[0m + \u001b[0;31mA__phi\u001b[0m*\u001b[0;34me_phi\u001b[0m\n",
"B = \u001b[0;31mB__rtheta\u001b[0m*\u001b[0;34me_r\u001b[0m^\u001b[0;34me_theta\u001b[0m + \u001b[0;31mB__rphi\u001b[0m*\u001b[0;34me_r\u001b[0m^\u001b[0;34me_phi\u001b[0m + \u001b[0;31mB__thetaphi\u001b[0m*\u001b[0;34me_theta\u001b[0m^\u001b[0;34me_phi\u001b[0m\n",
"grad*f = \u001b[0;36mD{r}\u001b[0;31mf\u001b[0m\u001b[0m*\u001b[0;34me_r\u001b[0m + \u001b[0;36mD{theta}\u001b[0;31mf\u001b[0m\u001b[0m*\u001b[0;34me_theta\u001b[0m/r + \u001b[0;36mD{phi}\u001b[0;31mf\u001b[0m\u001b[0m*\u001b[0;34me_phi\u001b[0m/(r*sin(theta))\n",
"grad|A = (r*\u001b[0;36mD{r}\u001b[0;31mA__r\u001b[0m\u001b[0m + 2*\u001b[0;31mA__r\u001b[0m + \u001b[0;31mA__theta\u001b[0m/tan(theta) + \u001b[0;36mD{theta}\u001b[0;31mA__theta\u001b[0m\u001b[0m + \u001b[0;36mD{phi}\u001b[0;31mA__phi\u001b[0m\u001b[0m/sin(theta))/r\n",
"-I*(grad^A) = (\u001b[0;31mA__phi\u001b[0m/tan(theta) + \u001b[0;36mD{theta}\u001b[0;31mA__phi\u001b[0m\u001b[0m - \u001b[0;36mD{phi}\u001b[0;31mA__theta\u001b[0m\u001b[0m/sin(theta))*\u001b[0;34me_r\u001b[0m/r + (-r*\u001b[0;36mD{r}\u001b[0;31mA__phi\u001b[0m\u001b[0m - \u001b[0;31mA__phi\u001b[0m + \u001b[0;36mD{phi}\u001b[0;31mA__r\u001b[0m\u001b[0m/sin(theta))*\u001b[0;34me_theta\u001b[0m/r + (r*\u001b[0;36mD{r}\u001b[0;31mA__theta\u001b[0m\u001b[0m + \u001b[0;31mA__theta\u001b[0m - \u001b[0;36mD{theta}\u001b[0;31mA__r\u001b[0m\u001b[0m)*\u001b[0;34me_phi\u001b[0m/r\n",
"grad^B = (r*\u001b[0;36mD{r}\u001b[0;31mB__thetaphi\u001b[0m\u001b[0m - \u001b[0;31mB__rphi\u001b[0m/tan(theta) + 2*\u001b[0;31mB__thetaphi\u001b[0m - \u001b[0;36mD{theta}\u001b[0;31mB__rphi\u001b[0m\u001b[0m + \u001b[0;36mD{phi}\u001b[0;31mB__rtheta\u001b[0m\u001b[0m/sin(theta))*\u001b[0;34me_r\u001b[0m^\u001b[0;34me_theta\u001b[0m^\u001b[0;34me_phi\u001b[0m/r\n",
"X = 1.2*\u001b[0;34me_x\u001b[0m + 2.34*\u001b[0;34me_y\u001b[0m + 0.555*\u001b[0;34me_z\u001b[0m\n",
"Nga(X,2) = 1.2*\u001b[0;34me_x\u001b[0m + 2.3*\u001b[0;34me_y\u001b[0m + 0.55*\u001b[0;34me_z\u001b[0m\n",
"X*Y = 12.7011 + 4.02078*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + 6.175185*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + 10.182*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"Nga(X*Y,2) = 13.0 + 4.0*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + 6.2*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + 10.0*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"g_{ij} = Matrix([\n",
"[ 0, (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m), (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)],\n",
"[(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m), 0, (\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)],\n",
"[(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m), (\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m), 1]])\n",
"(X^Y)**2 = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2\n",
"B = \u001b[0;34mX\u001b[0m^\u001b[0;34mY\u001b[0m - (\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*\u001b[0;34mX\u001b[0m^\u001b[0;34me\u001b[0m + (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*\u001b[0;34mY\u001b[0m^\u001b[0;34me\u001b[0m\n",
"B**2 = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2 - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)\n",
"#L = X^Y^e is a non-euclidian line\n",
"B = L*e = \u001b[0;34mX\u001b[0m^\u001b[0;34mY\u001b[0m - (\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*\u001b[0;34mX\u001b[0m^\u001b[0;34me\u001b[0m + (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*\u001b[0;34mY\u001b[0m^\u001b[0;34me\u001b[0m\n",
"B*e*B.rev() = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(-(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m) + 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m))*\u001b[0;34me\u001b[0m\n",
"B**2 = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*((\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m) - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m))\n",
"L**2 = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*((\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m) - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m))\n",
"s = sinh(alpha/2) and c = cosh(alpha/2)\n",
"exp(alpha*B/(2*|B|)) = c + (1/B)*s*\u001b[0;34mX\u001b[0m^\u001b[0;34mY\u001b[0m - (\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)*s*\u001b[0;34mX\u001b[0m^\u001b[0;34me\u001b[0m + (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)*s*\u001b[0;34mY\u001b[0m^\u001b[0;34me\u001b[0m\n",
"R*X*R.rev() = ((\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2*(1/B)**2*s**2 - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)**2*s**2 + 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(1/B)*c*s - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)*c*s + c**2)*\u001b[0;34mX\u001b[0m\n",
" + 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)**2*(1/B)*c*s*\u001b[0;34mY\u001b[0m\n",
" + 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)*s*(-(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(1/B)*s + 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)*s - c)*\u001b[0;34me\u001b[0m\n",
"Objective is to determine value of C = cosh(alpha) such that W = 0\n",
"Z|Y = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**3*(1/B)**2*s**2 - 4*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)**2*s**2 + 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2*(1/B)*c*s + 4*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)**2*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)**2*(1/B)**2*s**2 - 4*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*(1/B)*c*s + (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*c**2\n",
"S = sinh(alpha) and C = cosh(alpha)\n",
"W = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*C - (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*C + (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m) + S*sqrt((\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2 - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m))\n",
"Scalar Coefficient = (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)\n",
"Cosh Coefficient = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m) - (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)\n",
"Sinh Coefficient = sqrt((\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2 - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m))\n",
"|B| = sqrt((\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2 - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m))\n",
"Require a*C**2+b*C+c = 0\n",
"a = (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)**2*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)**2\n",
"b = 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)*((\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m) - (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m))\n",
"c = (\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)**2 - 2*(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)*(\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m) + (\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)**2*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)**2\n",
"cosh(alpha) = C = -b/(2*a) = -(\u001b[0;34mX\u001b[0m.\u001b[0;34mY\u001b[0m)/((\u001b[0;34mX\u001b[0m.\u001b[0;34me\u001b[0m)*(\u001b[0;34mY\u001b[0m.\u001b[0;34me\u001b[0m)) + 1\n",
"g_{ij} =\n",
" Matrix([\n",
"[1, 0, 0, 0, 0],\n",
"[0, 1, 0, 0, 0],\n",
"[0, 0, 1, 0, 0],\n",
"[0, 0, 0, 0, 2],\n",
"[0, 0, 0, 2, 0]])\n",
"F(a) = \u001b[0;34me_1\u001b[0m + \u001b[0;34mn\u001b[0m/2 - \u001b[0;34mnbar\u001b[0m/2\n",
"F(b) = \u001b[0;34me_2\u001b[0m + \u001b[0;34mn\u001b[0m/2 - \u001b[0;34mnbar\u001b[0m/2\n",
"F(c) = -\u001b[0;34me_1\u001b[0m + \u001b[0;34mn\u001b[0m/2 - \u001b[0;34mnbar\u001b[0m/2\n",
"F(d) = \u001b[0;34me_3\u001b[0m + \u001b[0;34mn\u001b[0m/2 - \u001b[0;34mnbar\u001b[0m/2\n",
"F(x) = x1*\u001b[0;34me_1\u001b[0m + x2*\u001b[0;34me_2\u001b[0m + x3*\u001b[0;34me_3\u001b[0m + (x1**2/2 + x2**2/2 + x3**2/2)*\u001b[0;34mn\u001b[0m - \u001b[0;34mnbar\u001b[0m/2\n",
"a = e1, b = e2, c = -e1, and d = e3\n",
"A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.\n",
"Circle through a, b, and c\n",
"Circle: A^B^C^X = 0 = -x3*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m + x3*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mnbar\u001b[0m + (x1**2/2 + x2**2/2 + x3**2/2 - 1/2)*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m\n",
"Line through a and b\n",
"Line : A^B^n^X = 0 = -x3*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m + (x1/2 + x2/2 - 1/2)*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m + x3*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m/2 - x3*\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m/2\n",
"Sphere through a, b, c, and d\n",
"Sphere: A^B^C^D^X = 0 = (-x1**2/2 - x2**2/2 - x3**2/2 + 1/2)*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m\n",
"Plane through a, b, and d\n",
"Plane : A^B^n^D^X = 0 = (-x1/2 - x2/2 - x3/2 + 1/2)*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m\n",
"Hyperbolic Circle: (A^B^e)^X = 0 = = -x3*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m\n",
" - x3*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mnbar\u001b[0m\n",
" + (-x1**2/2 + x1 - x2**2/2 + x2 - x3**2/2 - 1/2)*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m\n",
" + x3*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m\n",
" - x3*\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34mnbar\u001b[0m\n",
"g_{ij} =\n",
" Matrix([\n",
"[(\u001b[0;34mp1\u001b[0m.\u001b[0;34mp1\u001b[0m), (\u001b[0;34mp1\u001b[0m.\u001b[0;34mp2\u001b[0m), (\u001b[0;34mp1\u001b[0m.\u001b[0;34mp3\u001b[0m), 0, 0],\n",
"[(\u001b[0;34mp1\u001b[0m.\u001b[0;34mp2\u001b[0m), (\u001b[0;34mp2\u001b[0m.\u001b[0;34mp2\u001b[0m), (\u001b[0;34mp2\u001b[0m.\u001b[0;34mp3\u001b[0m), 0, 0],\n",
"[(\u001b[0;34mp1\u001b[0m.\u001b[0;34mp3\u001b[0m), (\u001b[0;34mp2\u001b[0m.\u001b[0;34mp3\u001b[0m), (\u001b[0;34mp3\u001b[0m.\u001b[0;34mp3\u001b[0m), 0, 0],\n",
"[ 0, 0, 0, 0, 2],\n",
"[ 0, 0, 0, 2, 0]])\n",
"Extracting direction of line from L = P1^P2^n\n",
"(L|n)|nbar = 2*\u001b[0;34mp1\u001b[0m - 2*\u001b[0;34mp2\u001b[0m\n",
"Extracting plane of circle from C = P1^P2^P3\n",
"((C^n)|n)|nbar = 2*\u001b[0;34mp1\u001b[0m^\u001b[0;34mp2\u001b[0m - 2*\u001b[0;34mp1\u001b[0m^\u001b[0;34mp3\u001b[0m + 2*\u001b[0;34mp2\u001b[0m^\u001b[0;34mp3\u001b[0m\n",
"(p2-p1)^(p3-p1) = \u001b[0;34mp1\u001b[0m^\u001b[0;34mp2\u001b[0m - \u001b[0;34mp1\u001b[0m^\u001b[0;34mp3\u001b[0m + \u001b[0;34mp2\u001b[0m^\u001b[0;34mp3\u001b[0m\n",
"g_{ij} =\n",
" Matrix([\n",
"[ 0, -1, (\u001b[0;34mP1\u001b[0m.\u001b[0;34ma\u001b[0m)],\n",
"[ -1, 0, (\u001b[0;34mP2\u001b[0m.\u001b[0;34ma\u001b[0m)],\n",
"[(\u001b[0;34mP1\u001b[0m.\u001b[0;34ma\u001b[0m), (\u001b[0;34mP2\u001b[0m.\u001b[0;34ma\u001b[0m), (\u001b[0;34ma\u001b[0m.\u001b[0;34ma\u001b[0m)]])\n",
"B**2 = 1\n",
"a' = a-(a^B)*B = -(\u001b[0;34mP2\u001b[0m.\u001b[0;34ma\u001b[0m)*\u001b[0;34mP1\u001b[0m - (\u001b[0;34mP1\u001b[0m.\u001b[0;34ma\u001b[0m)*\u001b[0;34mP2\u001b[0m\n",
"A+ = a'+a'*B = -2*(\u001b[0;34mP2\u001b[0m.\u001b[0;34ma\u001b[0m)*\u001b[0;34mP1\u001b[0m\n",
"A- = a'-a'*B = -2*(\u001b[0;34mP1\u001b[0m.\u001b[0;34ma\u001b[0m)*\u001b[0;34mP2\u001b[0m\n",
"(A+)^2 = 0\n",
"(A-)^2 = 0\n",
"a|B = -(\u001b[0;34mP2\u001b[0m.\u001b[0;34ma\u001b[0m)*\u001b[0;34mP1\u001b[0m + (\u001b[0;34mP1\u001b[0m.\u001b[0;34ma\u001b[0m)*\u001b[0;34mP2\u001b[0m\n",
"g_{ij} =\n",
" Matrix([\n",
"[ 1, (\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m), (\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m)],\n",
"[(\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m), 1, (\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m)],\n",
"[(\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m), (\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m), 1]])\n",
"E = \u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m\n",
"E**2 = (\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m)**2 - 2*(\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m)*(\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m)*(\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m) + (\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m)**2 + (\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m)**2 - 1\n",
"E1 = (e2^e3)*E = ((\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m)**2 - 1)*\u001b[0;34me1\u001b[0m + ((\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m) - (\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m)*(\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m))*\u001b[0;34me2\u001b[0m + (-(\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m)*(\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m) + (\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m))*\u001b[0;34me3\u001b[0m\n",
"E2 =-(e1^e3)*E = ((\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m) - (\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m)*(\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m))*\u001b[0;34me1\u001b[0m + ((\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m)**2 - 1)*\u001b[0;34me2\u001b[0m + (-(\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m)*(\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m) + (\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m))*\u001b[0;34me3\u001b[0m\n",
"E3 = (e1^e2)*E = (-(\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m)*(\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m) + (\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m))*\u001b[0;34me1\u001b[0m + (-(\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m)*(\u001b[0;34me1\u001b[0m.\u001b[0;34me3\u001b[0m) + (\u001b[0;34me2\u001b[0m.\u001b[0;34me3\u001b[0m))*\u001b[0;34me2\u001b[0m + ((\u001b[0;34me1\u001b[0m.\u001b[0;34me2\u001b[0m)**2 - 1)*\u001b[0;34me3\u001b[0m\n",
"E1|e2 = 0\n",
"E1|e3 = 0\n",
"E2|e1 = 0\n",
"E2|e3 = 0\n",
"E3|e1 = 0\n",
"E3|e2 = 0\n",
"(E1|e1)/E**2 = 1\n",
"(E2|e2)/E**2 = 1\n",
"(E3|e3)/E**2 = 1\n",
"e3d.g = Matrix([\n",
"[1, 0, 0],\n",
"[0, 1, 0],\n",
"[0, 0, 1]])\n",
"Signature = (3,0) I = \u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m I**2 = -1\n",
"e3d.g = Matrix([\n",
"[2, 0, 0],\n",
"[0, 2, 0],\n",
"[0, 0, 2]])\n",
"Signature = (3,0) I = sqrt(2)*\u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m/4 I**2 = -1\n",
"e3d.g = Matrix([\n",
"[1, 0, 0, 0],\n",
"[0, -1, 0, 0],\n",
"[0, 0, -1, 0],\n",
"[0, 0, 0, -1]])\n",
"Signature = (1,3) I = \u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m^\u001b[0;34me4\u001b[0m I**2 = -1\n",
"e3d.g = Matrix([\n",
"[2, 0, 0, 0],\n",
"[0, -2, 0, 0],\n",
"[0, 0, -2, 0],\n",
"[0, 0, 0, -2]])\n",
"Signature = (1,3) I = \u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m^\u001b[0;34me4\u001b[0m/4 I**2 = -1\n",
"e4d.g = Matrix([\n",
"[1, 0, 0, 0],\n",
"[0, 1, 0, 0],\n",
"[0, 0, 1, 0],\n",
"[0, 0, 0, 1]])\n",
"Signature = (4,0) I = \u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m^\u001b[0;34me4\u001b[0m I**2 = 1\n",
"cf4d.g = Matrix([\n",
"[1, 0, 0, 0, 0],\n",
"[0, 1, 0, 0, 0],\n",
"[0, 0, 1, 0, 0],\n",
"[0, 0, 0, 1, 0],\n",
"[0, 0, 0, 0, -1]])\n",
"Signature = (4,1) I = \u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m^\u001b[0;34me4\u001b[0m^\u001b[0;34me5\u001b[0m I**2 = -1\n",
"cf4d.g = Matrix([\n",
"[2, 0, 0, 0, 0],\n",
"[0, 2, 0, 0, 0],\n",
"[0, 0, 2, 0, 0],\n",
"[0, 0, 0, 2, 0],\n",
"[0, 0, 0, 0, -2]])\n",
"Signature = (4,1) I = sqrt(2)*\u001b[0;34me1\u001b[0m^\u001b[0;34me2\u001b[0m^\u001b[0;34me3\u001b[0m^\u001b[0;34me4\u001b[0m^\u001b[0;34me5\u001b[0m/8 I**2 = -1\n"
]
}
],
"source": [
"%run terminal_check.py"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"s**2\n",
"\u001b[0;31mf\u001b[0m\n",
"\u001b[0;36mD{s}\u001b[0;31mf\u001b[0m\u001b[0m\n",
"\\frac{d}{d s} f{\\left(s^{2} \\right)}\n"
]
}
],
"source": [
"%run 1d_grad.py"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"********************************************************************************\n",
"def coefs_test():\n",
" Print_Function()\n",
"\n",
" (o3d,e1,e2,e3) = Ga.build('e_1 e_2 e_3',g=[1,1,1])\n",
" print(o3d.blades.flat)\n",
" print(o3d.mv_blades.flat)\n",
" v = o3d.mv('v', 'vector')\n",
" print(v)\n",
" print(v.blade_coefs([e3,e1]))\n",
" A = o3d.mv('A', 'mv')\n",
" print(A)\n",
" print(A.blade_coefs([e1^e3,e3,e1^e2,e1^e2^e3]))\n",
" print(A.blade_coefs())\n",
" return\n",
"\n",
"Code output:\n",
"\n",
"(1, \u001b[0;34me_1\u001b[0m, \u001b[0;34me_2\u001b[0m, \u001b[0;34me_3\u001b[0m, \u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m, \u001b[0;34me_1\u001b[0m^\u001b[0;34me_3\u001b[0m, \u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m, \u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m)\n",
"(1, \u001b[0;34me_1\u001b[0m, \u001b[0;34me_2\u001b[0m, \u001b[0;34me_3\u001b[0m, \u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m, \u001b[0;34me_1\u001b[0m^\u001b[0;34me_3\u001b[0m, \u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m, \u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m)\n",
"v__1*\u001b[0;34me_1\u001b[0m + v__2*\u001b[0;34me_2\u001b[0m + v__3*\u001b[0;34me_3\u001b[0m\n",
"[v__3, v__1]\n",
"A + A__1*\u001b[0;34me_1\u001b[0m + A__2*\u001b[0;34me_2\u001b[0m + A__3*\u001b[0;34me_3\u001b[0m + A__12*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m + A__13*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_3\u001b[0m + A__23*\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m + A__123*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34me_3\u001b[0m\n",
"[A__13, A__3, A__12, A__123]\n",
"[A, A__1, A__2, A__3, A__12, A__13, A__23, A__123]\n"
]
}
],
"source": [
"%run coefs_test.py"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \u001b[0;34me_1\u001b[0m/2 + \u001b[0;34mn\u001b[0m/8 - \u001b[0;34m\\bar{n}\u001b[0m/2\n",
"B = 2*\u001b[0;34me_1\u001b[0m + 2*\u001b[0;34mn\u001b[0m - \u001b[0;34m\\bar{n}\u001b[0m/2\n",
"C = 4*\u001b[0;34me_1\u001b[0m/5 + 3*\u001b[0;34me_2\u001b[0m/5 + \u001b[0;34mn\u001b[0m/2 - \u001b[0;34m\\bar{n}\u001b[0m/2\n",
"D = 4*\u001b[0;34me_1\u001b[0m/5 - 3*\u001b[0;34me_2\u001b[0m/5 + \u001b[0;34mn\u001b[0m/2 - \u001b[0;34m\\bar{n}\u001b[0m/2\n",
"T = -9*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m/20 - 9*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34m\\bar{n}\u001b[0m/20 + 9*\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m^\u001b[0;34m\\bar{n}\u001b[0m/16\n",
"U = \u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m - \u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m^\u001b[0;34m\\bar{n}\u001b[0m\n",
"inter = -9*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m/5 + 9*\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m/8 - 9*\u001b[0;34me_2\u001b[0m^\u001b[0;34m\\bar{n}\u001b[0m/8\n",
"B = -9*\u001b[0;34me_1\u001b[0m^\u001b[0;34me_2\u001b[0m/5 + 9*\u001b[0;34me_2\u001b[0m^\u001b[0;34mn\u001b[0m/8 - 9*\u001b[0;34me_2\u001b[0m^\u001b[0;34m\\bar{n}\u001b[0m/8\n",
"B**2 = 729/400\n",
"NB = 27/20\n",
"20*\u001b[0;34me_1\u001b[0m/9 + 5*\u001b[0;34me_2\u001b[0m/3\n",
"20*\u001b[0;34me_1\u001b[0m/9 - 5*\u001b[0;34me_2\u001b[0m/3\n",
"One intersection point x = 20*\u001b[0;34me_1\u001b[0m/9 + 5*\u001b[0;34me_2\u001b[0m/3 + 25*\u001b[0;34mn\u001b[0m/18 - 25*\u001b[0;34m\\bar{n}\u001b[0m/18\n",
"The other intersection point y = 20*\u001b[0;34me_1\u001b[0m/9 - 5*\u001b[0;34me_2\u001b[0m/3 + 25*\u001b[0;34mn\u001b[0m/18 - 25*\u001b[0;34m\\bar{n}\u001b[0m/18\n",
"x**2 = 0\n",
"y**2 = 0\n",
"T^x = 0\n",
"T^y = 0\n",
"U^x = 0\n",
"U^y = 0\n"
]
}
],
"source": [
"%run ConformalConsole.py"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"u__x*\u001b[0;34me_x\u001b[0m + u__y*\u001b[0;34me_y\u001b[0m + u__z*\u001b[0;34me_z\u001b[0m\n",
"v__x*\u001b[0;34me_x\u001b[0m + v__y*\u001b[0;34me_y\u001b[0m + v__z*\u001b[0;34me_z\u001b[0m\n",
"(u__x*v__y - u__y*v__x)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m + (u__x*v__z - u__z*v__x)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + (u__y*v__z - u__z*v__y)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"True\n",
"exp(uv) = cos(sqrt(u__x**2*v__y**2 + u__x**2*v__z**2 - 2*u__x*u__y*v__x*v__y - 2*u__x*u__z*v__x*v__z + u__y**2*v__x**2 + u__y**2*v__z**2 - 2*u__y*u__z*v__y*v__z + u__z**2*v__x**2 + u__z**2*v__y**2)) + (u__x*v__y - u__y*v__x)*sin(sqrt(u__x**2*v__y**2 + u__x**2*v__z**2 - 2*u__x*u__y*v__x*v__y - 2*u__x*u__z*v__x*v__z + u__y**2*v__x**2 + u__y**2*v__z**2 - 2*u__y*u__z*v__y*v__z + u__z**2*v__x**2 + u__z**2*v__y**2))*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m/sqrt(u__x**2*v__y**2 + u__x**2*v__z**2 - 2*u__x*u__y*v__x*v__y - 2*u__x*u__z*v__x*v__z + u__y**2*v__x**2 + u__y**2*v__z**2 - 2*u__y*u__z*v__y*v__z + u__z**2*v__x**2 + u__z**2*v__y**2) + (u__x*v__z - u__z*v__x)*sin(sqrt(u__x**2*v__y**2 + u__x**2*v__z**2 - 2*u__x*u__y*v__x*v__y - 2*u__x*u__z*v__x*v__z + u__y**2*v__x**2 + u__y**2*v__z**2 - 2*u__y*u__z*v__y*v__z + u__z**2*v__x**2 + u__z**2*v__y**2))*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m/sqrt(u__x**2*v__y**2 + u__x**2*v__z**2 - 2*u__x*u__y*v__x*v__y - 2*u__x*u__z*v__x*v__z + u__y**2*v__x**2 + u__y**2*v__z**2 - 2*u__y*u__z*v__y*v__z + u__z**2*v__x**2 + u__z**2*v__y**2) + (u__y*v__z - u__z*v__y)*sin(sqrt(u__x**2*v__y**2 + u__x**2*v__z**2 - 2*u__x*u__y*v__x*v__y - 2*u__x*u__z*v__x*v__z + u__y**2*v__x**2 + u__y**2*v__z**2 - 2*u__y*u__z*v__y*v__z + u__z**2*v__x**2 + u__z**2*v__y**2))*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m/sqrt(u__x**2*v__y**2 + u__x**2*v__z**2 - 2*u__x*u__y*v__x*v__y - 2*u__x*u__z*v__x*v__z + u__y**2*v__x**2 + u__y**2*v__z**2 - 2*u__y*u__z*v__y*v__z + u__z**2*v__x**2 + u__z**2*v__y**2)\n"
]
}
],
"source": [
"%run exp_check.py"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Lt(\u001b[0;34me0\u001b[0m) = \u001b[0;34me0\u001b[0m + 2*\u001b[0;34me1\u001b[0m\n",
"Lt(\u001b[0;34me1\u001b[0m) = 3*\u001b[0;34me0\u001b[0m + 4*\u001b[0;34me1\u001b[0m\n",
"Lt(\u001b[0;34me0\u001b[0m) = \u001b[0;34me0\u001b[0m + 2*\u001b[0;34me1\u001b[0m\n",
"Lt(\u001b[0;34me1\u001b[0m) = 3*\u001b[0;34me0\u001b[0m + 4*\u001b[0;34me1\u001b[0m\n",
"Matrix([\n",
"[1, 2],\n",
"[3, 4]])\n"
]
}
],
"source": [
"%run lt_check.py"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Lt(\u001b[0;34me0\u001b[0m) = \u001b[0;34me0\u001b[0m + 2*\u001b[0;34me1\u001b[0m + 3*\u001b[0;34me2\u001b[0m\n",
"Lt(\u001b[0;34me1\u001b[0m) = 4*\u001b[0;34me0\u001b[0m + 5*\u001b[0;34me1\u001b[0m + 6*\u001b[0;34me2\u001b[0m\n",
"Lt(\u001b[0;34me2\u001b[0m) = 7*\u001b[0;34me0\u001b[0m + 8*\u001b[0;34me1\u001b[0m + 9*\u001b[0;34me2\u001b[0m\n",
"A.lt_dict[e0]= \u001b[0;34me0\u001b[0m + 2*\u001b[0;34me1\u001b[0m + 3*\u001b[0;34me2\u001b[0m\n",
"A.lt_dict[e1]= 4*\u001b[0;34me0\u001b[0m + 5*\u001b[0;34me1\u001b[0m + 6*\u001b[0;34me2\u001b[0m\n",
"A.lt_dict[e2]= 7*\u001b[0;34me0\u001b[0m + 8*\u001b[0;34me1\u001b[0m + 9*\u001b[0;34me2\u001b[0m\n",
"Matrix([\n",
"[1, 4, 7],\n",
"[2, 5, 8],\n",
"[3, 6, 9]])\n",
"v__x*\u001b[0;34me0\u001b[0m + v__y*\u001b[0;34me1\u001b[0m + v__z*\u001b[0;34me2\u001b[0m\n",
"(v__x + 4*v__y + 7*v__z)*\u001b[0;34me0\u001b[0m + (2*v__x + 5*v__y + 8*v__z)*\u001b[0;34me1\u001b[0m + (3*v__x + 6*v__y + 9*v__z)*\u001b[0;34me2\u001b[0m\n"
]
}
],
"source": [
"%run lt_test.py"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"********************************************************************************\n",
"def Mv_setup_options():\n",
" Print_Function()\n",
"\n",
" (o3d,e1,e2,e3) = Ga.build('e_1 e_2 e_3',g=[1,1,1])\n",
" v = o3d.mv('v', 'vector')\n",
" print(v)\n",
"\n",
" (o3d,e1,e2,e3) = Ga.build('e*1|2|3',g=[1,1,1])\n",
" v = o3d.mv('v', 'vector')\n",
" print(v)\n",
"\n",
" (o3d,e1,e2,e3) = Ga.build('e*x|y|z',g=[1,1,1])\n",
" v = o3d.mv('v', 'vector')\n",
" print(v)\n",
"\n",
" coords = symbols('x y z',real=True)\n",
" (o3d,e1,e2,e3) = Ga.build('e',g=[1,1,1],coords=coords)\n",
" v = o3d.mv('v', 'vector')\n",
" print(v)\n",
"\n",
" print(v.grade(2))\n",
" print(v.i_grade)\n",
"\n",
" return\n",
"\n",
"Code output:\n",
"\n",
"v__1*\u001b[0;34me_1\u001b[0m + v__2*\u001b[0;34me_2\u001b[0m + v__3*\u001b[0;34me_3\u001b[0m\n",
"v__1*\u001b[0;34me_1\u001b[0m + v__2*\u001b[0;34me_2\u001b[0m + v__3*\u001b[0;34me_3\u001b[0m\n",
"v__x*\u001b[0;34me_x\u001b[0m + v__y*\u001b[0;34me_y\u001b[0m + v__z*\u001b[0;34me_z\u001b[0m\n",
"v__x*\u001b[0;34me_x\u001b[0m + v__y*\u001b[0;34me_y\u001b[0m + v__z*\u001b[0;34me_z\u001b[0m\n",
"0\n",
"1\n"
]
}
],
"source": [
"%run mv_setup_options.py"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = z*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m - y*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + x*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"grad^A = 3*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"\n",
"f = 1/sqrt(x**2 + y**2 + z**2)\n",
"grad*f = -x*\u001b[0;34me_x\u001b[0m/(x**2 + y**2 + z**2)**(3/2) - y*\u001b[0;34me_y\u001b[0m/(x**2 + y**2 + z**2)**(3/2) - z*\u001b[0;34me_z\u001b[0m/(x**2 + y**2 + z**2)**(3/2)\n",
"\n",
"B = z*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m/sqrt(x**2 + y**2 + z**2) - y*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m/sqrt(x**2 + y**2 + z**2) + x*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m/sqrt(x**2 + y**2 + z**2)\n",
"\n",
"grad^B = 2*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m/sqrt(x**2 + y**2 + z**2)\n"
]
}
],
"source": [
"%run prob_not_solenoidal.py"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Matrix([\n",
"[ \u001b[0;36mD{u}\u001b[0;31mx\u001b[0m\u001b[0m**2 + \u001b[0;36mD{u}\u001b[0;31my\u001b[0m\u001b[0m**2 + \u001b[0;36mD{u}\u001b[0;31mz\u001b[0m\u001b[0m**2, \u001b[0;36mD{u}\u001b[0;31mx\u001b[0m\u001b[0m*\u001b[0;36mD{v}\u001b[0;31mx\u001b[0m\u001b[0m + \u001b[0;36mD{u}\u001b[0;31my\u001b[0m\u001b[0m*\u001b[0;36mD{v}\u001b[0;31my\u001b[0m\u001b[0m + \u001b[0;36mD{u}\u001b[0;31mz\u001b[0m\u001b[0m*\u001b[0;36mD{v}\u001b[0;31mz\u001b[0m\u001b[0m],\n",
"[\u001b[0;36mD{u}\u001b[0;31mx\u001b[0m\u001b[0m*\u001b[0;36mD{v}\u001b[0;31mx\u001b[0m\u001b[0m + \u001b[0;36mD{u}\u001b[0;31my\u001b[0m\u001b[0m*\u001b[0;36mD{v}\u001b[0;31my\u001b[0m\u001b[0m + \u001b[0;36mD{u}\u001b[0;31mz\u001b[0m\u001b[0m*\u001b[0;36mD{v}\u001b[0;31mz\u001b[0m\u001b[0m, \u001b[0;36mD{v}\u001b[0;31mx\u001b[0m\u001b[0m**2 + \u001b[0;36mD{v}\u001b[0;31my\u001b[0m\u001b[0m**2 + \u001b[0;36mD{v}\u001b[0;31mz\u001b[0m\u001b[0m**2]])\n",
"1\n",
"0\n",
"0\n",
"1\n"
]
}
],
"source": [
"%run reciprocal.py"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a = a__x*\u001b[0;34me_x\u001b[0m + a__y*\u001b[0;34me_y\u001b[0m + a__z*\u001b[0;34me_z\u001b[0m\n",
"b = b__x*\u001b[0;34me_x\u001b[0m + b__y*\u001b[0;34me_y\u001b[0m + b__z*\u001b[0;34me_z\u001b[0m\n",
"c = c__x*\u001b[0;34me_x\u001b[0m + c__y*\u001b[0;34me_y\u001b[0m + c__z*\u001b[0;34me_z\u001b[0m\n",
"ab = (a__x - b__x)*\u001b[0;34me_x\u001b[0m + (a__y - b__y)*\u001b[0;34me_y\u001b[0m + (a__z - b__z)*\u001b[0;34me_z\u001b[0m\n",
"ab/|ab| = (a__x - b__x)*\u001b[0;34me_x\u001b[0m/ab_mag + (a__y - b__y)*\u001b[0;34me_y\u001b[0m/ab_mag + (a__z - b__z)*\u001b[0;34me_z\u001b[0m/ab_mag\n",
"R_ab = cos(theta_ab/2) + (a__z - b__z)*cos(theta_ab/2)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m/ab_mag + (-a__y + b__y)*cos(theta_ab/2)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m/ab_mag + (a__x - b__x)*cos(theta_ab/2)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m/ab_mag\n",
"R_ab_rev = cos(theta_ab/2) + (-a__z + b__z)*cos(theta_ab/2)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m/ab_mag + (a__y - b__y)*cos(theta_ab/2)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m/ab_mag + (-a__x + b__x)*cos(theta_ab/2)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m/ab_mag\n",
"e_ab_x = (a__x**2 - 2*a__x*b__x - a__y**2 + 2*a__y*b__y - a__z**2 + 2*a__z*b__z + ab_mag**2 + b__x**2 - b__y**2 - b__z**2)*cos(theta_ab/2)**2*\u001b[0;34me_x\u001b[0m/ab_mag**2 + 2*(a__x*a__y - a__x*b__y - a__y*b__x + ab_mag*(-a__z + b__z) + b__x*b__y)*cos(theta_ab/2)**2*\u001b[0;34me_y\u001b[0m/ab_mag**2 + 2*(a__x*a__z - a__x*b__z - a__z*b__x + ab_mag*(a__y - b__y) + b__x*b__z)*cos(theta_ab/2)**2*\u001b[0;34me_z\u001b[0m/ab_mag**2\n",
"e_ab_y = 2*(a__x*a__y - a__x*b__y - a__y*b__x + ab_mag*(a__z - b__z) + b__x*b__y)*cos(theta_ab/2)**2*\u001b[0;34me_x\u001b[0m/ab_mag**2 + (-a__x**2 + 2*a__x*b__x + a__y**2 - 2*a__y*b__y - a__z**2 + 2*a__z*b__z + ab_mag**2 - b__x**2 + b__y**2 - b__z**2)*cos(theta_ab/2)**2*\u001b[0;34me_y\u001b[0m/ab_mag**2 + 2*(a__y*a__z - a__y*b__z - a__z*b__y + ab_mag*(-a__x + b__x) + b__y*b__z)*cos(theta_ab/2)**2*\u001b[0;34me_z\u001b[0m/ab_mag**2\n",
"e_ab_z = 2*(a__x*a__z - a__x*b__z - a__z*b__x + ab_mag*(-a__y + b__y) + b__x*b__z)*cos(theta_ab/2)**2*\u001b[0;34me_x\u001b[0m/ab_mag**2 + 2*(a__y*a__z - a__y*b__z - a__z*b__y + ab_mag*(a__x - b__x) + b__y*b__z)*cos(theta_ab/2)**2*\u001b[0;34me_y\u001b[0m/ab_mag**2 + (-a__x**2 + 2*a__x*b__x - a__y**2 + 2*a__y*b__y + a__z**2 - 2*a__z*b__z + ab_mag**2 - b__x**2 - b__y**2 + b__z**2)*cos(theta_ab/2)**2*\u001b[0;34me_z\u001b[0m/ab_mag**2\n",
"cos(phi/2) - sin(phi/2)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m\n",
"cos(phi/2) + sin(phi/2)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_y\u001b[0m\n",
"-sin(phi)*\u001b[0;34me_x\u001b[0m + cos(phi)*\u001b[0;34me_y\u001b[0m\n",
"cos(theta/2) - sin(theta/2)*cos(phi)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m - sin(phi)*sin(theta/2)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"cos(theta/2) + sin(theta/2)*cos(phi)*\u001b[0;34me_x\u001b[0m^\u001b[0;34me_z\u001b[0m + sin(phi)*sin(theta/2)*\u001b[0;34me_y\u001b[0m^\u001b[0;34me_z\u001b[0m\n",
"e_r = cos(phi)*cos(theta)*\u001b[0;34me_x\u001b[0m + sin(phi)*cos(theta)*\u001b[0;34me_y\u001b[0m + sin(theta)*\u001b[0;34me_z\u001b[0m\n",
"e_th = -sin(theta)*cos(phi)*\u001b[0;34me_x\u001b[0m - sin(phi)*sin(theta)*\u001b[0;34me_y\u001b[0m + cos(theta)*\u001b[0;34me_z\u001b[0m\n",
"e_phi = -sin(phi)*\u001b[0;34me_x\u001b[0m + cos(phi)*\u001b[0;34me_y\u001b[0m\n"
]
}
],
"source": [
"%run rotations.py"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}