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    "# Laplacian, Hessian, and Jacobian\n",
    "\n",
    "All three operators are assembled from the same array of partial\n",
    "derivatives.  For a scalar $f$ and vector map $F$,\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": [
    "## Coordinates and test fields\n",
    "\n",
    "We use $f=x^2+y^2+z^2$ and\n",
    "$F=(x^2,y^2,z^2)$.  Both have diagonal derivative matrices, which\n",
    "makes the expected answers easy to inspect.\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": {},
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    }
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   "source": [
    "∂/∂ scalarField parameters\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ffe321e",
   "metadata": {},
   "source": [
    "## Hessian and Laplacian\n",
    "\n",
    "`generateTensor` supplies the two free matrix indices.  Taking the\n",
    "trace of the resulting Hessian implements the Euclidean Laplacian.\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": [
    "## Jacobian matrix and determinant\n",
    "\n",
    "The same indexed construction differentiates each component of\n",
    "$F$.  Its determinant measures the local volume scaling.\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": [
    "The outputs are $2I_3$, $6$, and $8xyz$, respectively.  They show\n",
    "how tensor generation, contraction, and determinant calculation\n",
    "express three familiar multivariable operators in a uniform way.\n"
   ]
  }
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