Multilinear form

In abstract algebra and multilinear algebra, a multilinear form on V {displaystyle V} is a map of the typeIf ω {displaystyle omega } is a smooth ( n − 1 ) {displaystyle (n-1)} -form on an open set A ⊂ R m {displaystyle Asubset mathbf {R} ^{m}} and C {displaystyle C} is a smooth n {displaystyle n} -chain in A {displaystyle A} , then ∫ C d ω = ∫ ∂ C ω {displaystyle int _{C}domega =int _{partial C}omega } . In abstract algebra and multilinear algebra, a multilinear form on V {displaystyle V} is a map of the type where V {displaystyle V} is a vector space over the field K {displaystyle K} (or more generally, a module over a commutative ring), that is separately K-linear in each of its k {displaystyle k} arguments. (The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.) A multilinear k-form on V {displaystyle V} over R {displaystyle mathbf {R} } is called a (covariant) k-tensor, and the vector space of such forms is usually denoted T k ( V ) {displaystyle {mathcal {T}}^{k}(V)} or L k ( V ) {displaystyle {mathcal {L}}^{k}(V)} . Given k-tensor f ∈ T k ( V ) {displaystyle fin {mathcal {T}}^{k}(V)} and ℓ-tensor g ∈ T ℓ ( V ) {displaystyle gin {mathcal {T}}^{ell }(V)} , a product f ⊗ g ∈ T k + ℓ ( V ) {displaystyle fotimes gin {mathcal {T}}^{k+ell }(V)} , known as the tensor product, can be defined by the property for all v 1 , … , v k + ℓ ∈ V {displaystyle v_{1},ldots ,v_{k+ell }in V} . The tensor product of multilinear forms is not commutative; however it is bilinear and associative: