Discrete Contraction and Inner Product

et and be two 1-forms. Their inner product can be expressed as

where represents the pairing between a vector field and a one form. In this context one can think of vector fields as row matrices, and forms as row matrices.

The sharp operator is a mapping from forms to vector fields, and the flat operator is its inverse.

Contraction with a one form

The contraction with a one form is simply given by

Contraction with a two form

Suppose the two form can be expressed as the wedge product of two one forms such as . Then