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   "source": [
    "# Exterior Derivative and $d^2=0$\n",
    "\n",
    "The exterior derivative sends a $k$-form to a $(k+1)$-form.  On a\n",
    "scalar function it is the gradient one-form,\n",
    "\n",
    "$$df=\\frac{\\partial f}{\\partial x^i},dx^i,$$\n",
    "\n",
    "and on all differential forms it satisfies the fundamental identity\n",
    "$d^2=0$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5c4b7c9",
   "metadata": {},
   "source": [
    "## A coordinate-free implementation pattern\n",
    "\n",
    "Egison's tensor derivative can map the partial-derivative operator\n",
    "across the coordinate vector.  The polymorphic definition below works\n",
    "for a scalar or a tensor-valued expression.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "72085ad7",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol x, y, z : MathValue\n",
    "\n",
    "def params : Vector MathValue := [| x, y, z |]\n",
    "\n",
    "def d {a} (X : a) : DiffForm a := !(flip ∂/∂) params X\n",
    "\n",
    "def f : MathValue := x ^ 2 + y ^ 2 + z ^ 2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49250bca",
   "metadata": {},
   "source": [
    "For $f=x^2+y^2+z^2$, the first exterior derivative is the radial\n",
    "gradient one-form $2x\\,dx+2y\\,dy+2z\\,dz$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "aae6d940",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 2 x \\\\ 2 y \\\\ 2 z\\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d f\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6bcc4b2",
   "metadata": {},
   "source": [
    "Applying the tensor derivative a second time first produces the raw\n",
    "Hessian.  This intermediate object has not yet been projected onto its\n",
    "alternating differential-form part.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a695fd86",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 2 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d (d f)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd2ec05c",
   "metadata": {},
   "source": [
    "The Hessian of a smooth scalar is symmetric, while a two-form is\n",
    "antisymmetric.  Their alternating projection therefore vanishes.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c9b3e356",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dfNormalize (d (d f))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6ed78ee",
   "metadata": {},
   "source": [
    "This computation is the coordinate expression of $d^2f=0$: mixed\n",
    "partial derivatives cancel pairwise after antisymmetrization.  Showing\n",
    "both the Hessian and its normalized form separates ordinary tensor\n",
    "differentiation from the geometric exterior derivative.\n"
   ]
  }
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