Wedge Product and Antisymmetrization

Differential forms multiply with the alternating wedge product. For one-forms $a$ and $b$,

$$(a\wedge b)_{ij}=\frac12(a_i b_j-a_j b_i),$$

so $a\wedge b=-b\wedge a$ and $a\wedge a=0$. Egison first constructs an indexed tensor product and then exposes the alternating differential-form representative through dfNormalize.

Coordinate one-forms in $\mathbb R^3$

In the ordered basis $(dx,dy,dz)$, each basis one-form is represented by a vector. The ambient dimension and coordinates are included to make the geometric setting explicit.

declare symbol x, y, z : MathValue

def N : Integer := 3
def params : Vector MathValue := [| x, y, z |]

def dx : DiffForm Integer := [| 1, 0, 0 |]
def dy : DiffForm Integer := [| 0, 1, 0 |]
def dz : DiffForm Integer := [| 0, 0, 1 |]

Raw indexed product

The raw wedge expression retains the ordered component generated by the tensor operation. Looking at it before normalization makes the representation convention visible.

dx ∧ dy
$\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$

Alternating two-form

Normalization distributes that component over the antisymmetric matrix. The entries at $(1,2)$ and $(2,1)$ have opposite signs and carry the conventional factor $1/2$.

dfNormalize (dx ∧ dy)
$\begin{pmatrix} 0 & \frac{1}{2} & 0 \\ \frac{-1}{2} & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$

Antisymmetrization also removes a repeated basis direction. Although the intermediate indexed product $dz\wedge dz$ has a diagonal entry, its differential-form normalization is zero.

dfNormalize (dz ∧ dz)
$\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$

Thus dfNormalize is not cosmetic: it projects an indexed tensor onto its alternating part. Once normalized, the displayed tensors obey the geometric identities $dx\wedge dy=-dy\wedge dx$ and $dz\wedge dz=0$.

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