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    "# Hodge Star in Euclidean Three-Space\n",
    "\n",
    "A metric and an orientation identify $k$-forms with $(3-k)$-forms.\n",
    "In oriented Euclidean coordinates,\n",
    "\n",
    "$$\\star dx=dy\\wedge dz,\\qquad\n",
    "  \\star(dx\\wedge dy)=dz.$$\n",
    "\n",
    "The Hodge star is where the metric enters differential-form calculus.\n"
   ]
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    "## Euclidean metric and basis forms\n",
    "\n",
    "We use the identity metric on $(x,y,z)$ and the standard orientation.\n",
    "The metric is typed as a matrix of symbolic mathematical values so the\n",
    "same definition pattern also works for nonconstant metrics.\n"
   ]
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   "source": [
    "declare symbol x, y, z : MathValue\n",
    "\n",
    "def N : Integer := 3\n",
    "def params : Vector MathValue := [| x, y, z |]\n",
    "def g : Matrix MathValue :=\n",
    "  [| [| 1, 0, 0 |], [| 0, 1, 0 |], [| 0, 0, 1 |] |]\n",
    "\n",
    "def dx : DiffForm MathValue := [| 1, 0, 0 |]\n",
    "def dy : DiffForm MathValue := [| 0, 1, 0 |]\n",
    "def dz : DiffForm MathValue := [| 0, 0, 1 |]\n"
   ]
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   "source": [
    "## Definition of the Hodge star\n",
    "\n",
    "For a $k$-form $A$, the Levi-Civita tensor supplies the complementary\n",
    "indices, inverse metrics raise the contracted indices, and\n",
    "$\\sqrt{|\\det g|}$ supplies the volume density.\n"
   ]
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   "source": [
    "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"
   ]
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   "source": [
    "A one-form becomes a two-form supported in the complementary oriented\n",
    "plane.\n"
   ]
  },
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   "id": "5118d224",
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       "$\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}$"
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   "source": [
    "hodge dx\n"
   ]
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   "source": [
    "Conversely, the oriented area form in the $xy$-plane becomes the\n",
    "one-form normal to that plane.\n"
   ]
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       "$\\begin{pmatrix} 0 \\\\ 0 \\\\ 1\\\\ \\end{pmatrix}$"
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   "source": [
    "hodge (wedge dx dy)\n"
   ]
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   "id": "23226d48",
   "metadata": {},
   "source": [
    "The two outputs are the component versions of\n",
    "$\\star dx=dy\\wedge dz$ and $\\star(dx\\wedge dy)=dz$.  In three-dimensional\n",
    "Euclidean signature, applying $\\star$ twice returns a one-form or a\n",
    "two-form with positive sign; changing the metric signature changes that\n",
    "sign, as the Minkowski-space notebook demonstrates.\n"
   ]
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