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{{nota disambigua|la forma sesquilineare simmetrica definita positiva|Forma sesquilineare#Prodotto interno|Prodotto interno}}
In [[matematica]], il '''prodotto interno''' o '''derivata interna''' è una [[derivazione]] di [[algebra graduata|grado]] −1 sull'[[algebra esterna]] delle [[forma differenziale|forme differenziali]] su [[varietà differenziabile|varietà lisce]].
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==Definition==
The interior product is defined to be the [[tensor contraction|contraction]] of a [[differential form]] with a [[vector field]]. Thus if ''X'' is a vector field on the [[manifold]] ''M'', then
:<math>\iota_X\colon \Omega^p(M) \to \Omega^{p-1}(M)</math>
is the [[Map (mathematics)|map]] which sends a ''p''-form ''ω'' to the (''p''−1)-form ''ι''<sub>''X''</sub>''ω'' defined by the property that
:<math>( \iota_X\omega )(X_1,\ldots,X_{p-1})=\omega(X,X_1,\ldots,X_{p-1})</math>
for any vector fields ''X''<sub>1</sub>, ..., ''X''<sub>''p''−1</sub>.
The interior product is the unique [[derivation (algebra)|antiderivation]] of degree −1 on the [[exterior algebra]] such that on one-forms ''α''
:<math>\displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle</math>,
where <,> is the [[duality pairing]] between ''α'' and the vector ''X''. Explicitly, if ''β'' is a ''p''-form and γ is a ''q''-form, then
:<math> \iota_X(\beta\wedge\gamma) = (\iota_X\beta)\wedge\gamma+(-1)^p\beta\wedge(\iota_X\gamma). </math>
The above relation says that the interior product obeys a graded [[Product rule|Leibniz rule]]. An operation equipped with linearity and a Leibniz rule is often called a derivative.
==Properties==
By antisymmetry of forms,
:<math> \iota_X \iota_Y \omega = - \iota_Y \iota_X \omega </math>
and so <math> \iota_X^2 = 0 </math>. This may be compared to the [[exterior derivative]] ''d'', which has the property {{nowrap|1=''d''<sup>2</sup> = 0}}. The interior product relates the [[exterior derivative]] and [[Lie derivative]] of differential forms by '''''Cartan's identity''''':
:<math> \mathcal L_X\omega = \mathrm d (\iota_X \omega) + \iota_X \mathrm d\omega. </math>
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 <math>X</math>, <math>Y</math> satisfies the identity
:<math> \iota_{[X,Y]}= \left[ \mathcal{L}_X, \iota_Y \right] . </math>
==See also==
* [[Cap product]]
* [[Inner product]]
* [[Tensor contraction]]
==References==
{{reflist}}
*Theodore Frankel, ''The Geometry of Physics: An Introduction''; Cambridge University Press, 3rd ed. 2011
{{DEFAULTSORT:Interior Product}}
[[Category:Differential forms]]
[[Category:Multilinear algebra]]-->
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