WedgeProduct - crowlogic/arb4j GitHub Wiki

The wedge product, also known as the exterior product, is an antisymmetric, bilinear operation that combines vectors or differential forms in the context of exterior algebra. Given two vectors $u$ and $v$ in a vector space $V$, their wedge product $u \wedge v$ is an element in the exterior algebra $\Lambda(V)$.

The wedge product has some important properties:

1. Antisymmetry: For any two elements $u$ and $v$ in $V$, their wedge product satisfies: $$u \wedge v = - (v \wedge u)$$

2. Bilinearity: The wedge product is linear in each argument, so for any scalar $a$, $b \in F$ and vectors $u$, $v$, $w \in V$, we have:

$$(a * u + b * v) \wedge w = a * (u \wedge w) + b * (v \wedge w)$$

3. Graded associativity: Although the wedge product is not associative in general, the exterior algebra still obeys a modified form of associativity called graded associativity:

$$(u \wedge v) \wedge w = u \wedge (v \wedge w)$$

for any $u$, $v$, and $w$ in the exterior algebra.

The wedge product plays a crucial role in differential geometry and is used in the study of differential forms, which are essential for defining concepts like integration, Stokes' theorem, and de Rham cohomology.