Definition:Wedge Product

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Definition

Let $\alpha$ and $\beta$ be two differential forms.

Let $\alpha$ be an $x$-form.

Let $\beta$ be a $y$-form.


The wedge product $\alpha \wedge \beta$ is defined as the linear antisymmetric map from $F^x \times F^y \to F^{x + y}$, where $F^a$ is the set of $a$-forms in some manifold.


Let $x_0$ be a specific point in a manifold $X$.

Let $\alpha$ be an $x$-form.

Let $\phi$ be a $1$-form.

Let there be a set of vectors $\mathbf u_1, \mathbf u_2, \dotsc, \mathbf u_x, \mathbf v \in \map {T_{x_0} } X$.


The wedge product is defined as:

$\alpha \wedge \map \phi {\mathbf u_1, \mathbf u_2, \dotsc, \mathbf u_x, \mathbf v} := \sum_P \map \varepsilon P \, \map \alpha {P_1} \, \map \phi {P_2}$

where:

$P$ is some permutation of $\mathbf u_1, \mathbf u_2, \dots, \mathbf u_x, \mathbf v$
$P_1$ is the first $x$ terms of the permutation $P$
$P_2$ the final term of permutation $P$
$\varepsilon$ is the permutation symbol of $P$.

The sum is taken over all possible permutations.


This definition extends to wedge products of arbitrary forms through the linearity and antisymmetric conditions.