# Discrete Contraction and Inner ProductΒΆ

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