Contraction in Coordinates

Given a manifold $\text{[math]}$ , the interior product is defined as the contraction of a differential form with a vector field.

The contraction of a one-form $\text{[math]}$ with a vector field $\text{[math]}$ is given by

The contraction of a two-form $\text{[math]}$ with the same vector field is given by

The contraction of a three-form $\text{[math]}$ is going to be

In 2D the explicit formula for the contractions are

If 3D, on the other hand, the contractions become