information:filtering
Interior product

Suppose that V {\displaystyle V} is finite-dimensional. If V ∗ {\displaystyle V^{*}} denotes the dual space to the vector space V {\displaystyle V}, then for each α ∈ V ∗ {\displaystyle \alpha \in V^{*}}, it is possible to define an antiderivation on the algebra ⋀ (V) {\displaystyle {\textstyle \bigwedge }(V)},

This derivation is called the interior product with α {\displaystyle \alpha }, or sometimes the insertion operator, or contraction by α {\displaystyle \alpha }.

Suppose that w ∈ ⋀ k (V) {\displaystyle w\in {\textstyle \bigwedge }^{\!k}(V)}. Then w {\displaystyle w} is a multilinear mapping of V ∗ {\displaystyle V^{*}} to K {\displaystyle K}, so it is defined by its values on the k -fold Cartesian product V ∗ × V ∗ × ⋯ × V ∗ {\displaystyle V^{*}\times V^{*}\times \dots \times V^{*}}. If u, u,..., u are k − 1 {\displaystyle k-1} elements of V ∗ {\displaystyle V^{*}}, then define

Additionally, let ι α f = 0 {\displaystyle \iota _{\alpha }f=0} whenever f {\displaystyle f} is a pure scalar (i.e., belonging to ⋀ 0 (V) {\displaystyle {\textstyle \bigwedge }^{\!0}(V)}).