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   "source": [
    "# Hodge Laplacian in Polar Coordinates\n",
    "\n",
    "This notebook derives the scalar Hodge Laplacian from the metric and\n",
    "differential-form operators.  For polar coordinates $(r,\\theta)$ with\n",
    "$r>0$,\n",
    "\n",
    "$$g=dr^2+r^2d\\theta^2.$$\n",
    "\n",
    "With the codifferential convention used below, the result is the\n",
    "negative of the usual positive-coordinate Laplace operator.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a37b734",
   "metadata": {},
   "source": [
    "## Polar metric\n",
    "\n",
    "Both the covariant and contravariant metrics carry explicit tensor\n",
    "indices.  This lets Egison contract them automatically in the Hodge\n",
    "formula.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1a4aa201",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol r, θ : MathValue\n",
    "\n",
    "def N : Integer := 2\n",
    "def x : Vector MathValue := [| r, θ |]\n",
    "\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  [| [| 1, 0 |], [| 0, r ^ 2 |] |]_i_j\n",
    "def g~i~j : Matrix MathValue :=\n",
    "  [| [| 1, 0 |], [| 0, r ^ (-2) |] |]~i~j\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "8cab770d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 1 & 0 \\\\ 0 & r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
     },
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     "output_type": "display_data"
    }
   ],
   "source": [
    "g_#_#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9508b31a",
   "metadata": {},
   "source": [
    "## Exterior derivative and Hodge star\n",
    "\n",
    "The Hodge star combines the Levi-Civita tensor, the inverse metric,\n",
    "and the volume density $\\sqrt{|g|}=r$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "6aa28359",
   "metadata": {},
   "outputs": [],
   "source": [
    "def d (A : Tensor MathValue) : Tensor MathValue :=\n",
    "  !(flip ∂/∂) x A\n",
    "\n",
    "def hodge (A : Tensor MathValue) : Tensor MathValue :=\n",
    "  let k := dfOrder A\n",
    "   in withSymbols [i, j]\n",
    "        sqrt (M.det g_#_#) *\n",
    "        foldl\n",
    "          (.)\n",
    "          ((subrefs A (map 1#j_$1 (between 1 k))) .\n",
    "           (subrefs (ε' N k) (map 1#i_$1 (between 1 N))))\n",
    "          (map 1#g~(i_$1)~(j_$1) [1..k])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5893c721",
   "metadata": {},
   "source": [
    "## Codifferential and Laplacian\n",
    "\n",
    "The codifferential is the metric adjoint of $d$ and can be written in\n",
    "terms of two Hodge stars.  The degree test selects the appropriate\n",
    "endpoint formula for zero- and top-degree forms.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "faddc387",
   "metadata": {},
   "outputs": [],
   "source": [
    "def δ (A : Tensor MathValue) : Tensor MathValue :=\n",
    "  let k := dfOrder A\n",
    "   in (-1) ^ (N * (k + 1) + 1) * (hodge (d (hodge A)))\n",
    "\n",
    "def Δ (A : Tensor MathValue) : Tensor 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)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8840f8f2",
   "metadata": {},
   "source": [
    "Applying the operator to an arbitrary scalar function leaves its\n",
    "derivatives symbolic, so the coordinate formula is visible directly.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "04b501b1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$-\\frac{\\partial^2 f}{\\partial 1^2} - \\frac{\\partial f}{\\partial 1} r^{-1} - \\frac{\\partial^2 f}{\\partial 2^2} r^{-2}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Δ f\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf201951",
   "metadata": {},
   "source": [
    "The displayed expression is\n",
    "\n",
    "$$\n",
    "\\Delta f=-\\left(\n",
    "  \\frac{\\partial^2f}{\\partial r^2}\n",
    "  +\\frac1r\\frac{\\partial f}{\\partial r}\n",
    "  +\\frac1{r^2}\\frac{\\partial^2f}{\\partial\\theta^2}\n",
    "\\right).\n",
    "$$\n",
    "\n",
    "The $1/r$ and $1/r^2$ terms are not inserted by hand: they arise from\n",
    "the determinant and inverse metric inside the Hodge star.\n"
   ]
  }
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