In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, 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
 is sometimes written as  [1]
[1]
Definition
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
 is a vector field on the manifold  then
 then 
 is the map which sends a
is the map which sends a  -form
-form  to the
 to the  -form
-form  defined by the property that
 defined by the property that
 for any vector fields
for any vector fields  
When  is a scalar field (0-form),
 is a scalar field (0-form),  by convention.
 by convention.
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms  
 where
where  is the duality pairing between
 is the duality pairing between  and the vector
 and the vector  Explicitly, if
  Explicitly, if  is a
 is a  -form and
-form and  is a
 is a  -form, then
-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.
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
If in local coordinates  the vector field
 the vector field  is given by
 is given by
 
then the interior product is given by
 where
where  is the form obtained by omitting
 is the form obtained by omitting  from
 from  .
.
By antisymmetry of forms, 
 and so
and so  This may be compared to the exterior derivative
 This may be compared to the exterior derivative  which has the property
 which has the property  
The interior product with respect to the commutator of two vector fields  
  satisfies the identity
 satisfies the identity
![{\displaystyle \iota _{[X,Y]}=\left[{\mathcal {L}}_{X},\iota _{Y}\right]=\left[\iota _{X},{\mathcal {L}}_{Y}\right].}](./_assets_/bbbe7d287098ef3452dc8ffd37be92eef823d3e5.svg) Proof. For any k-form
Proof. For any k-form  ,
, ![{\displaystyle {\mathcal {L}}_{X}(\iota _{Y}\Omega )-\iota _{Y}({\mathcal {L}}_{X}\Omega )=({\mathcal {L}}_{X}\Omega )(Y,-)+\Omega ({\mathcal {L}}_{X}Y,-)-({\mathcal {L}}_{X}\Omega )(Y,-)=\iota _{{\mathcal {L}}_{X}Y}\Omega =\iota _{[X,Y]}\Omega }](./_assets_/fb63909db7da6a9aad430ccd1eff3d8a2965f27a.svg) and similarly for the other result.
and similarly for the other result.
Cartan identity
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]
Proof by direct computation [5]
Since vector fields are locally integrable, we can always find a local coordinate system  such that the vector field
 such that the vector field  corresponds to the partial derivative with respect to the first coordinate, i.e.,
 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  -forms. There are only two cases:
-forms. There are only two cases:
Case 1:   . Direct computation yields:
. Direct computation yields: 
Case 2:   . Direct computation yields:
. Direct computation yields: 
 
See also
Notes
References
- Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011
- Loring W. Tu, An Introduction to Manifolds, 2e, Springer. 2011. doi:10.1007/978-1-4419-7400-6
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