Interior product

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 "ι""X""ω" is sometimes written as .
Definition.
The interior product is defined to be the contraction of a differential form with a vector field. Thus if "X" is a vector field on the manifold "M", then 
is the map which sends a "p"-form "ω" to the ("p"−1)-form "ι""X""ω" defined by the property that
for any vector fields "X"1, ..., "X""p"−1.
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms "α"
where is the duality pairing between "α" and the vector "X". Explicitly, if "β" is a "p"-form and "γ" is a q-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 formula_6. This may be compared to the exterior derivative "d", which has the property .
The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (a.k.a. Cartan identity, Cartan homotopy formula or Cartan magic formula):
This identity defines a duality between the exterior and interior derivatives. Cartan's identity is important in symplectic geometry and general relativity: see moment map. The Cartan homotopy formula is named after Élie Cartan. 
The interior product with respect to the commutator of two vector fields formula_8, formula_9 satisfies the identity