Definition:Wedge Product

The wedge product $$\alpha \wedge \beta \ $$ of two 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.