Spatprodukt

Spatprodukt, or exterior product, is a binary operation in exterior algebra that takes two vectors from a vector space V over a field and returns a bivector in the second exterior power Λ^2(V). It is bilinear and alternating, meaning a ∧ b = - b ∧ a for any vectors a and b, and a ∧ a = 0. The operation extends to higher degrees, so the wedge product can be applied to any number of vectors to produce elements of Λ^k(V).

In coordinates, if V has a basis {e1, …, en} and vectors a = (a1, …, an), b = (b1, …, bn), then a ∧ b = sum over i<j of (ai bj − aj bi) (ei ∧ ej). The set {ei ∧ ej | i < j} forms a basis for Λ^2(V), and the dimension of Λ^2(V) is n(n−1)/2. The wedge product is antisymmetric and distributes over addition, and it is associative within the exterior algebra.

Geometrically, the Spatprodukt of two vectors encodes the oriented area of the parallelogram spanned by them; its magnitude equals that area, and its orientation is determined by the chosen orientation of V. In three dimensions with an inner product, the Hodge star maps bivectors to vectors via *(a ∧ b) = a × b, establishing a link between the wedge product and the cross product.

The exterior (Spatprodukt) generalizes to higher grades, forming the exterior algebra Λ(V) = ⊕k Λ^k(V). It is foundational in differential geometry, algebra, and physics for describing oriented volumes and forms.