1  {
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  7     "outputs": [
  8      {
  9       "data": {
 10        "text/latex": [
 11         "$\\displaystyle \\DeclareMathOperator{\\Tr}{Tr}$$\n",
 12         "$$\\DeclareMathOperator{\\Adj}{Adj}$$\n",
 13         "$$\\newcommand{\\bfrac}[2]{\\displaystyle\\frac{#1}{#2}}$$\n",
 14         "$$\\newcommand{\\lp}{\\left (}$$\n",
 15         "$$\\newcommand{\\rp}{\\right )}$$\n",
 16         "$$\\newcommand{\\paren}[1]{\\lp {#1} \\rp}$$\n",
 17         "$$\\newcommand{\\half}{\\frac{1}{2}}$$\n",
 18         "$$\\newcommand{\\llt}{\\left <}$$\n",
 19         "$$\\newcommand{\\rgt}{\\right >}$$\n",
 20         "$$\\newcommand{\\abs}[1]{\\left |{#1}\\right | }$$\n",
 21         "$$\\newcommand{\\pdiff}[2]{\\bfrac{\\partial {#1}}{\\partial {#2}}}$$\n",
 22         "$$\\newcommand{\\npdiff}[3]{\\bfrac{\\partial^{#3} {#1}}{\\partial {#2}^{#3}}}$$\n",
 23         "$$\\newcommand{\\lbrc}{\\left \\{}$$\n",
 24         "$$\\newcommand{\\rbrc}{\\right \\}}$$\n",
 25         "$$\\newcommand{\\W}{\\wedge}$$\n",
 26         "$$\\newcommand{\\prm}[1]{{#1}^{\\prime}}$$\n",
 27         "$$\\newcommand{\\ddt}[1]{\\bfrac{d{#1}}{dt}}$$\n",
 28         "$$\\newcommand{\\R}{\\dagger}$$\n",
 29         "$$\\newcommand{\\deriv}[3]{\\bfrac{d^{#3}#1}{d{#2}^{#3}}}$$\n",
 30         "$$\\newcommand{\\grade}[2]{\\left < {#1} \\right >_{#2}}$$\n",
 31         "$$\\newcommand{\\f}[2]{{#1}\\lp {#2} \\rp}$$\n",
 32         "$$\\newcommand{\\eval}[2]{\\left . {#1} \\right |_{#2}}$$\n",
 33         "$$\\newcommand{\\bs}[1]{\\boldsymbol{#1}}$$\n",
 34         "$$\\newcommand{\\grad}{\\bs{\\nabla}}$"
 35        ],
 36        "text/plain": [
 37         "<IPython.core.display.Math object>"
 38        ]
 39       },
 40       "metadata": {},
 41       "output_type": "display_data"
 42      }
 43     ],
 44     "source": [
 45      "# A geometric algebra for the unit sphere in R^3 \n",
 46      "# as a submanifold of R^3 with spherical coordintes.\n",
 47      "\n",
 48      "# Make SymPy available to this program:\n",
 49      "import sympy \n",
 50      "from sympy import *\n",
 51      "\n",
 52      "# Make GAlgebra available to this program:\n",
 53      "from galgebra.ga import *  \n",
 54      "from galgebra.mv import *\n",
 55      "from galgebra.printer import Fmt, GaPrinter, Format\n",
 56      "    # Fmt:       sets the way that a multivector's basis expansion is output.\n",
 57      "    # GaPrinter: makes GA output a little more readable.\n",
 58      "    # Format:    turns on latex printer.\n",
 59      "from galgebra.gprinter import gFormat, gprint\n",
 60      "gFormat()"
 61     ]
 62    },
 63    {
 64     "cell_type": "code",
 65     "execution_count": 2,
 66     "metadata": {},
 67     "outputs": [],
 68     "source": [
 69      "# A geometric algebra for the unit sphere in R^3 \n",
 70      "# as a submanifold of R^3 with spherical coordintes.\n",
 71      "\n",
 72      "# sp3: Base manifold\n",
 73      "sp3coords = (r, phi, theta) = symbols('r phi theta', real=True) \n",
 74      "sp3 = Ga('e', g=None, coords=sp3coords, \\\n",
 75      "    X=[r*sin(phi)*cos(theta), r*sin(phi)*sin(theta), r*cos(phi)], norm=True)\n",
 76      "(er, ephi, etheta) = sp3.mv()\n",
 77      "\n",
 78      "# sp2: Submanifold\n",
 79      "sp2coords = (p,t) = symbols('phi theta', real=True) # they output as Greek phi and theta\n",
 80      "  # Parameterize the unit sphere using the spherical coordinates of sp3:\n",
 81      "sp2param = [1, p, t] \n",
 82      "  # Map the sp3 coordinates of the sphere to its sp2 coordinates:\n",
 83      "sp2 = sp3.sm(sp2param, sp2coords, norm=True)\n",
 84      "\n",
 85      "(ep, et) = sp2.mv()\n",
 86      "(rp, rt) = sp2.mvr()\n",
 87      "\n",
 88      "# Derivatives\n",
 89      "grad  = sp2.grad\n",
 90      "from galgebra.dop import *\n",
 91      "pdph = Pdop(p)\n",
 92      "pdth = Pdop(t)"
 93     ]
 94    },
 95    {
 96     "cell_type": "code",
 97     "execution_count": 3,
 98     "metadata": {},
 99     "outputs": [
100      {
101       "data": {
102        "text/latex": [
103         "$\\displaystyle  \\boldsymbol{e}_{\\phi } \\frac{\\partial}{\\partial \\phi } + \\boldsymbol{e}_{\\theta } \\frac{1}{\\sin{\\left (\\phi  \\right )}} \\frac{\\partial}{\\partial \\theta } $"
104        ],
105        "text/plain": [
106         "<IPython.core.display.Math object>"
107        ]
108       },
109       "metadata": {},
110       "output_type": "display_data"
111      }
112     ],
113     "source": [
114      "gprint(grad) "
115     ]
116    },
117    {
118     "cell_type": "code",
119     "execution_count": null,
120     "metadata": {},
121     "outputs": [],
122     "source": []
123    }
124   ],
125   "metadata": {
126    "kernelspec": {
127     "display_name": "Python 3 (ipykernel)",
128     "language": "python",
129     "name": "python3"
130    },
131    "language_info": {
132     "codemirror_mode": {
133      "name": "ipython",
134      "version": 3
135     },
136     "file_extension": ".py",
137     "mimetype": "text/x-python",
138     "name": "python",
139     "nbconvert_exporter": "python",
140     "pygments_lexer": "ipython3",
141     "version": "3.11.8"
142    }
143   },
144   "nbformat": 4,
145   "nbformat_minor": 4
146  }