How To Do Differential Form Interior Product?

The interior product, also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation, is a degree-1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. It is a mapping from a p form ω to a (p – 1) form, as one argument of the p form is fixed to be a particular vector field, such as X. As a result, ιXω can only be evaluated at some vector fields.

The interior product is a dual notion of the wedge product in an exterior algebra LambdaV, where V is a vector space. Given an orthonormal basis (e_i) of V, the interior product is a dual notion of the wedge product. The interior product relates the exterior derivative and Lie derivative of differential forms by Cartan’s identity.

An r-form is a linear function with r slots for vectors that changes sign if any two slots are interchanged. An interior product with differential forms is a mathematical operation that combines a vector field and a differential form to produce a new differential form. The interior product of a form with a vector field is called i(X)ω, which is linear with respect to functions.

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The interior product of a form with a vector field just shoves the vector field into the first slot of evaluation in the form evaluated at some vector fields.