In mathematics, 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. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product is sometimes written as [1]
is the map which sends a -form to the -form defined by the property that
for any vector fields
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms
where is the duality pairing between and the vector Explicitly, if is a -form and is a -form, then
The above relation says that the interior product obeys a graded Leibniz rule. An operation satisfying linearity and a Leibniz rule is called a derivation.
Properties[]
By antisymmetry of forms,
and so This may be compared to the exterior derivative which has the property
The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula[2] or Cartan magic formula):