Related constructions and facts

Exterior forms are usually simply called “forms,” but as we saw before this term is also used to describe general multilinear mappings from vector spaces to scalars. Here we revisit these mappings to give their equivalent definitions in “tensor language.”

  • Bilinear form: a covariant tensor of order 2
  • Multilinear form: a covariant tensor of arbitrary order
  • Quadratic form: a symmetric covariant tensor of order 2
  • Algebraic form: a symmetric covariant tensor of arbitrary order
  • Exterior form: an anti-symmetric covariant tensor of arbitrary order
  • Pseudo inner product: a nondegenerate completely symmetric covariant tensor of order 2
  • Symplectic form: a nondegenerate completely anti-symmetric covariant tensor of order 2

The operation of the exterior product by a fixed 1-form \({\left(\varphi\wedge\right)}\) can be viewed as a linear mapping from \({\Lambda^{k}V^{*}}\) to \({\Lambda^{k+1}V^{*}}\). We can form a mapping that goes in the opposite direction, the interior product of \({\varphi}\) by a vector \({v}\), denoted \({i_{v}\varphi}\) (also denoted \({v\lrcorner\varphi}\)). This operation fixes the first vector argument of a \({k}\)-form, i.e. \({\left(i_{v}\varphi\right)\left(w_{2},\dotsc,w_{k}\right)\equiv\varphi\left(v,w_{2},\dotsc,w_{k}\right)}\). The interior product is thus a linear mapping from \({\Lambda^{k}V^{*}}\) to \({\Lambda^{k-1}V^{*}}\). We define \({i_{v}f\equiv0}\) for a 0-form \({f}\) and note that \({i_{v}\Omega=*(v^{\flat})}\).

Δ The relationship between the interior and exterior product is not that of “opposites,” i.e. neither reverses the effect of the other. Instead, as we will see, the interior product acts as a kind of derivative.

An Illustrated Handbook