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.
[1] The interior product is defined to be the contraction of a differential form with a vector field.
is a vector field on the manifold
is the map which sends a
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms
is the duality pairing between
( β ∧ γ ) =
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.
If in local coordinates
is the form obtained by omitting
This may be compared to the exterior derivative
The interior product with respect to the commutator of two vector fields
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):
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.
[3] The Cartan homotopy formula is named after Élie Cartan.
[4] Since vector fields are locally integrable, we can always find a local coordinate system
corresponds to the partial derivative with respect to the first coordinate, i.e.,
By linearity of the interior product, exterior derivative, and Lie derivative, it suffices to prove the Cartan's magic formula for monomial
Direct computation yields:
Direct computation yields:
is an anti-derivation on the exterior algebra.
Similarly, the interior product
On the other hand, the Lie derivative
The anti-commutator of two anti-derivations is a derivation.
To show that two derivations on the exterior algebra are equal, it suffices to show that they agree on a set of generators.
Locally, the exterior algebra is generated by 0-forms (smooth functions
Verify Cartan's magic formula on these two cases.