{
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  {
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   "source": [
    "# Wedge Product and Antisymmetrization\n",
    "\n",
    "Differential forms multiply with the alternating wedge product.\n",
    "For one-forms $a$ and $b$,\n",
    "\n",
    "$$(a\\wedge b)_{ij}=\\frac12(a_i b_j-a_j b_i),$$\n",
    "\n",
    "so $a\\wedge b=-b\\wedge a$ and $a\\wedge a=0$.  Egison first\n",
    "constructs an indexed tensor product and then exposes the alternating\n",
    "differential-form representative through `dfNormalize`.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c777e4a",
   "metadata": {},
   "source": [
    "## Coordinate one-forms in $\\mathbb R^3$\n",
    "\n",
    "In the ordered basis $(dx,dy,dz)$, each basis one-form is represented\n",
    "by a vector.  The ambient dimension and coordinates are included to\n",
    "make the geometric setting explicit.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "f0bc9356",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol x, y, z : MathValue\n",
    "\n",
    "def N : Integer := 3\n",
    "def params : Vector MathValue := [| x, y, z |]\n",
    "\n",
    "def dx : DiffForm Integer := [| 1, 0, 0 |]\n",
    "def dy : DiffForm Integer := [| 0, 1, 0 |]\n",
    "def dz : DiffForm Integer := [| 0, 0, 1 |]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a872c8c",
   "metadata": {},
   "source": [
    "## Raw indexed product\n",
    "\n",
    "The raw wedge expression retains the ordered component generated by\n",
    "the tensor operation.  Looking at it before normalization makes the\n",
    "representation convention visible.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "5d3cef2e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dx ∧ dy\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92429ee5",
   "metadata": {},
   "source": [
    "## Alternating two-form\n",
    "\n",
    "Normalization distributes that component over the antisymmetric\n",
    "matrix.  The entries at $(1,2)$ and $(2,1)$ have opposite signs and\n",
    "carry the conventional factor $1/2$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "3eba21e0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & \\frac{1}{2} & 0 \\\\ \\frac{-1}{2} & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dfNormalize (dx ∧ dy)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf2b9fb6",
   "metadata": {},
   "source": [
    "Antisymmetrization also removes a repeated basis direction.  Although\n",
    "the intermediate indexed product $dz\\wedge dz$ has a diagonal entry,\n",
    "its differential-form normalization is zero.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3b03c11a",
   "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 (dz ∧ dz)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0e171c2",
   "metadata": {},
   "source": [
    "Thus `dfNormalize` is not cosmetic: it projects an indexed tensor onto\n",
    "its alternating part.  Once normalized, the displayed tensors obey the\n",
    "geometric identities $dx\\wedge dy=-dy\\wedge dx$ and $dz\\wedge dz=0$.\n"
   ]
  }
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