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Template:Distinguish In mathematics, the interior product is a degree −1 antiderivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, is also called interior or inner multiplication, or the inner derivative or derivation, but should not be confused with an inner product. The interior product ι_Xω is sometimes written as X PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

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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

${\iota }_X:{\mathrm{\Omega }}^p(M)\to {\mathrm{\Omega }}^{p-1}(M)$

is the map which sends a p-form ω to the (p−1)-form ι_Xω defined by the property that

$({\iota }_X\omega )(X_1,\dots ,X_{p-1})=\omega (X,X_1,\dots ,X_{p-1})$

for any vector fields X₁,..., X_p−1.

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α

${\iota }_X\alpha =\alpha (X)=⟨\alpha ,X⟩$,

the duality pairing between α and the vector X. Explicitly, if β is a p-form and γ is a q-form, then

${\iota }_X(\beta \wedge \gamma )=({\iota }_X\beta )\wedge \gamma +(-1{)}^p\beta \wedge ({\iota }_X\gamma ).$

The above relation says that the interior product obeys a graded Leibniz rule. An operation equipped with linearity and a Leibniz rule is often called a derivative. The interior product is also known as the interior derivative.

Properties

By antisymmetry of forms,

${\iota }_X{\iota }_Y\omega =-{\iota }_Y{\iota }_X^{}\omega$

and so ${\iota }_X^2=0$. This may be compared to the exterior derivative d which has the property d² = 0. The interior product relates the exterior derivative and Lie derivative of differential forms by Cartan's identity:

${\mathcal{ℒ}}_X\omega =\mathrm{d}({\iota }_X\omega )+{\iota }_X\mathrm{d}\omega .$

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 interior product with respect to the commutator of two vector fields $X,Y$ satisfies the identity

${\iota }_{[X,Y]}={\mathcal{ℒ}}_X{\iota }_Y-{\iota }_Y{\mathcal{ℒ}}_X.$

See also

• Inner product
• Tensor contraction

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