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]


The interior product is defined to be the contraction of a differential form with a vector field. Thus if is a vector field on the manifold then

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.


If in local coordinates the vector field is given by

then the interior product is given by

where is the form obtained by omitting from .

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):

where the anticommutator was used. 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]

The interior product with respect to the commutator of two vector fields satisfies the identity

See also


  1. ^ The character ⨼ is U+2A3C INTERIOR PRODUCT in Unicode
  2. ^ Tu, Sec 20.5.
  3. ^ There is another formula called "Cartan formula". See Steenrod algebra.
  4. ^ Is "Cartan's magic formula" due to Élie or Henri?, MathOverflow, 2010-09-21, retrieved 2018-06-25