Interior product

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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]

Definition

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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  

When   is a scalar field (0-form),   by convention.

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

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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 with respect to the commutator of two vector fields     satisfies the identity  Proof. For any k-form  ,  and similarly for the other result.

Cartan identity

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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   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:

Case 1:  . Direct computation yields: 

Case 2:  . Direct computation yields: 

Proof by abstract algebra, credited to Shiing-Shen Chern[4]

The exterior derivative   is an anti-derivation on the exterior algebra. Similarly, the interior product   with a vector field   is also an anti-derivation. On the other hand, the Lie derivative   is a derivation.

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  ) and their differentials, exact 1-forms ( ). Verify Cartan's magic formula on these two cases.

See also

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  • Cap product – Method in algebraic topology
  • Inner product – Generalization of the dot product; used to define Hilbert spaces
  • Tensor contraction – Operation in mathematics and physics

Notes

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  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. ^ a b Is "Cartan's magic formula" due to Élie or Henri?, MathOverflow, 2010-09-21, retrieved 2018-06-25
  5. ^ Elementary Proof of the Cartan Magic Formula, Oleg Zubelevich

References

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  • 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