Definition:Wedge Product

Definition

The wedge product $\alpha \wedge \beta$ of two differential forms $\alpha, \beta$, where $\alpha$ is an $x$-form and $\beta$ a $y$-form, 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.

Given a specific point $x_0$ in a manifold $X$, an $x$-form $\alpha$, a $1$-form $\phi$, and a set of vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_x, \mathbf{v} \in T_{x_0}(X)$, the wedge product:

$\alpha \wedge \phi \left({\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_x, \mathbf{v} }\right) = \sum_P \varepsilon(P) \alpha(P_1) \phi(P_2)$

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 $P$, and $\varepsilon$ 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.