{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "82fca0f8",
   "metadata": {},
   "source": [
    "# 極座標のホッジラプラシアン\n",
    "\n",
    "このNotebookでは、計量と微分形式の演算子からスカラーに対するホッジラプラシアンを\n",
    "導出します。$r>0$ の極座標 $(r,\\theta)$ では、\n",
    "\n",
    "$$g=dr^2+r^2d\\theta^2.$$\n",
    "\n",
    "となります。以下で採用する余微分の規約では、結果は通常の座標表示による\n",
    "正符号のラプラス作用素に負号を付けたものになります。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a37b734",
   "metadata": {},
   "source": [
    "## 極座標の計量\n",
    "\n",
    "共変計量と反変計量の両方にテンソルの添字を明示します。これによりEgisonは、\n",
    "ホッジスターの公式に現れる縮約を自動的に処理できます。\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": [
       "$\\begin{pmatrix} 1 & 0 \\\\ 0 & r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g_#_#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9508b31a",
   "metadata": {},
   "source": [
    "## 外微分とホッジスター\n",
    "\n",
    "ホッジスターは、レヴィ・チヴィタテンソル、逆計量、体積密度\n",
    "$\\sqrt{|g|}=r$ を組み合わせて定義されます。\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": [
    "## 余微分とラプラシアン\n",
    "\n",
    "余微分は $d$ の計量に関する随伴であり、二つのホッジスターを用いて表せます。\n",
    "形式の次数を調べることで、0形式と最高次形式に適した端点の公式を選択します。\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": [
    "任意のスカラー関数に作用素を適用し、導関数を記号のまま残すことで、\n",
    "座標表示の公式を直接確認できます。\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": [
    "表示される式は\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",
    "です。$1/r$ と $1/r^2$ の項を手作業で加えているわけではありません。\n",
    "これらはホッジスターに含まれる計量の行列式と逆計量から現れます。\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Egison",
   "language": "egison",
   "name": "egison"
  },
  "language_info": {
   "codemirror_mode": "egison",
   "file_extension": ".egi",
   "mimetype": "text/x-egison",
   "name": "egison"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}