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    "# Hodge Laplacian in Spherical Coordinates\n",
    "\n",
    "In $(r,\\theta,\\phi)$ coordinates, the Euclidean metric is\n",
    "\n",
    "$$ds^2=dr^2+r^2d\\theta^2+r^2\\sin^2\\theta\\,d\\phi^2.$$\n",
    "\n",
    "We build $d$, $\\star$, and the codifferential from this metric.  With\n",
    "the codifferential convention used below, the scalar result is the\n",
    "negative of the usual positive-coordinate Laplace operator.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80c07a49",
   "metadata": {},
   "source": [
    "## Spherical metric\n",
    "\n",
    "The determinant is $r^4\\sin^2\\theta$, while the inverse metric contains\n",
    "the angular scale factors $r^{-2}$ and\n",
    "$(r^2\\sin^2\\theta)^{-1}$.\n"
   ]
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  {
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   "id": "8851b573",
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   "source": [
    "declare symbol r, θ, φ : MathValue\n",
    "\n",
    "def N : Integer := 3\n",
    "def x : Vector MathValue := [| r, θ, φ |]\n",
    "\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  [| [| 1, 0, 0 |]\n",
    "   , [| 0, r ^ 2, 0 |]\n",
    "   , [| 0, 0, r ^ 2 * (sin θ) ^ 2 |] |]_i_j\n",
    "def g~i~j : Matrix MathValue :=\n",
    "  [| [| 1, 0, 0 |]\n",
    "   , [| 0, 1 / r ^ 2, 0 |]\n",
    "   , [| 0, 0, 1 / (r ^ 2 * (sin θ) ^ 2) |] |]~i~j\n"
   ]
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   "execution_count": 2,
   "id": "2015dd74",
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       "$\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & r^{2} & 0 \\\\ 0 & 0 & \\sin(θ)^{2} r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
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   "source": [
    "g_#_#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c00a1543",
   "metadata": {},
   "source": [
    "## Differential-form operators\n",
    "\n",
    "The Hodge star raises the form indices with $g^{ij}$ and contracts them\n",
    "against the three-dimensional Levi-Civita tensor.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "210a0cf0",
   "metadata": {},
   "outputs": [],
   "source": [
    "def d (A : Tensor MathValue) : Tensor MathValue :=\n",
    "  !(flip ∂/∂) x A\n",
    "\n",
    "def hodge (A : DiffForm MathValue) : DiffForm MathValue :=\n",
    "  let k := dfOrder A\n",
    "   in withSymbols [i, j]\n",
    "        sqrt (abs (M.det g_#_#)) *\n",
    "        foldl\n",
    "          (.)\n",
    "          ((ε' N k)_(i_1)..._(i_N) . A..._(j_1)..._(j_k))\n",
    "          (map (\\n -> g~(i_n)~(j_n)) [1..k])\n",
    "\n",
    "def δ (A : DiffForm MathValue) : DiffForm MathValue :=\n",
    "  let k := dfOrder A\n",
    "   in ((-1) ^ (N * (k + 1) + 1)) * hodge (d (hodge A))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08fa7851",
   "metadata": {},
   "source": [
    "The Hodge Laplacian is $d\\delta+\\delta d$, with shorter endpoint\n",
    "formulas for scalars and volume forms.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "138d5b01",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Δ (A : DiffForm MathValue) : DiffForm MathValue :=\n",
    "  match dfOrder A as integer with\n",
    "  | #0 -> δ (d A)\n",
    "  | #N -> d (δ A)\n",
    "  | _  -> d (δ A) + δ (d A)\n",
    "\n",
    "def f : MathValue := function (r, θ, φ)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5dd385fe",
   "metadata": {},
   "outputs": [
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     "data": {
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       "$-\\frac{\\partial^2 f}{\\partial 1^2} - 2 \\frac{\\partial f}{\\partial 1} r^{-1} - \\frac{\\partial^2 f}{\\partial 2^2} r^{-2} - \\cos(θ) \\sin(θ)^{-1} \\frac{\\partial f}{\\partial 2} r^{-2} - \\sin(θ)^{-2} \\frac{\\partial^2 f}{\\partial 3^2} r^{-2}$"
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   "source": [
    "Δ f\n"
   ]
  },
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   "source": [
    "After simplification, the output is\n",
    "\n",
    "$$\n",
    "\\Delta f=-\\left(\n",
    "f_{rr}+\\frac{2}{r}f_r\n",
    "+\\frac1{r^2}f_{\\theta\\theta}\n",
    "+\\frac{\\cos\\theta}{r^2\\sin\\theta}f_\\theta\n",
    "+\\frac1{r^2\\sin^2\\theta}f_{\\phi\\phi}\n",
    "\\right).\n",
    "$$\n",
    "\n",
    "Each coordinate-dependent coefficient follows from the metric; the\n",
    "calculation contains no spherical-coordinate Laplacian formula as an\n",
    "input.\n"
   ]
  }
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