wedgeprodukt

WedgeProdukt is the term used in some contexts for what is more widely known as the wedge product in exterior algebra. It is a binary operation defined on a vector space V over a field F, extended to the entire exterior algebra Λ(V). The operation maps a k-vector and an l-vector to a (k+l)-vector and is typically denoted by the wedge sign ∧.

The wedgeProdukt is bilinear, associative, and anti-commutative in a graded sense. For vectors u and v in V, u ∧ v is an element of Λ^2(V) and satisfies u ∧ v = − v ∧ u, with u ∧ u = 0. More generally, if α ∈ Λ^k(V) and β ∈ Λ^l(V), then α ∧ β ∈ Λ^{k+l}(V) and α ∧ β = (−1)^{kl} β ∧ α. The operation distributes over addition in each argument.

Basis and geometry. In a basis {e_i} of V, the space Λ^2(V) has basis elements e_i ∧ e_j with i < j, reflecting antisymmetry. A simple bivector a ∧ b encodes an oriented parallelogram spanned by a and b; its magnitude relates to the oriented area. Higher-degree wedges encode oriented k-volumes.

Applications and context. The wedgeProdukt is central in differential geometry and the calculus of differential forms. It underpins constructions such as the exterior algebra Λ(V), the exterior derivative, and the formulation of volume and orientation in manifolds. It also appears in physics, computer graphics, and geometry to express oriented areas, volumes, and multi-vector quantities.

See also. Exterior algebra; differential forms; exterior derivative; Hodge dual.