{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "a83a0f93",
   "metadata": {},
   "source": [
    "# ラプラシアン・ヘッセ行列・ヤコビ行列\n",
    "\n",
    "これら3つの演算は、いずれも同じ偏微分の配列から組み立てられます。\n",
    "スカラー関数$f$とベクトル値写像$F$に対して、\n",
    "\n",
    "$$\n",
    "(\\operatorname{Hess}f)_{ij}=\\partial_i\\partial_j f,\n",
    "\\qquad\n",
    "\\Delta f=\\operatorname{tr}(\\operatorname{Hess}f),\n",
    "\\qquad\n",
    "J_F=(\\partial_jF_i)_{ij}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b41b8969",
   "metadata": {},
   "source": [
    "## 座標と検算用の場\n",
    "\n",
    "$f=x^2+y^2+z^2$と$F=(x^2,y^2,z^2)$を用います。どちらも\n",
    "微分行列が対角行列になるため、期待する結果を容易に確認できます。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3a52a9c1",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol x, y, z : MathValue\n",
    "\n",
    "def parameters : Vector MathValue := [| x, y, z |]\n",
    "def scalarField : MathValue := x^2 + y^2 + z^2\n",
    "def vectorField : Vector MathValue := [| x^2, y^2, z^2 |]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1b333b03",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 2 x \\\\ 2 y \\\\ 2 z\\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "∂/∂ scalarField parameters\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ffe321e",
   "metadata": {},
   "source": [
    "## ヘッセ行列とラプラシアン\n",
    "\n",
    "`generateTensor`で行列の2つの自由添字を与えます。得られた\n",
    "ヘッセ行列のトレースを取ることで、ユークリッド空間の\n",
    "ラプラシアンを実装します。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b62505e2",
   "metadata": {},
   "outputs": [],
   "source": [
    "def hessian (f : MathValue) : Matrix MathValue :=\n",
    "  generateTensor\n",
    "    (\\[a, b] -> ∂/∂ (∂/∂ f parameters_a) parameters_b)\n",
    "    [3, 3]\n",
    "\n",
    "def laplacian (f : MathValue) : MathValue := trace (hessian f)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7cdbfa34",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 2 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hessian scalarField\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "6a0c4f8d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$6$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "laplacian scalarField\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb0ad2fa",
   "metadata": {},
   "source": [
    "## ヤコビ行列と行列式\n",
    "\n",
    "同じ添字付きの構成を使って、$F$の各成分を微分します。\n",
    "その行列式は、局所的な体積の拡大率を表します。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "6fd7c37f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def jacobian (v : Vector MathValue) : Matrix MathValue :=\n",
    "  generateTensor\n",
    "    (\\[a, b] -> ∂/∂ v_a parameters_b)\n",
    "    [3, 3]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "10abc531",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 2 x & 0 & 0 \\\\ 0 & 2 y & 0 \\\\ 0 & 0 & 2 z \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "jacobian vectorField\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "406a4283",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$8 x y z$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M.det (jacobian vectorField)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b90604d5",
   "metadata": {},
   "source": [
    "出力はそれぞれ$2I_3$、$6$、$8xyz$です。テンソルの生成、縮約、\n",
    "行列式の計算を組み合わせることで、多変数解析でおなじみの3つの演算を\n",
    "統一的に表現できることが分かります。\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
}