|
|
| Line 1: |
Line 1: |
| {{distinguish|Inner product}}
| | I am Celeste from Amsterdam. I am learning to play the Xylophone. Other hobbies are Knapping.<br><br>Here is my web-site - Fifa 15 Coin Generator; [http://Web38.Wwwsrv03.t3-network.de/guestbook/guestbook.php web38.wwwsrv03.t3-Network.de], |
| In [[mathematics]], the '''interior product''' is a [[graded algebra|degree]] −1 [[antiderivation]] on the [[exterior algebra]] of [[differential form]]s 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''<sub>''X''</sub>''ω'' is sometimes written as ''X'' {{Unicode|⨼}} ''ω''; this character is U+2A3C in [[Unicode]].
| |
| | |
| ==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>, | |
| 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. The interior product is also known as the interior 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 ''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,Y</math> satisfies the identity
| |
| | |
| :<math> \iota_{[X,Y]}=\mathcal L_X \iota_Y-\iota_Y \mathcal L_X. </math>
| |
| | |
| ==See also==
| |
| * [[Inner product]]
| |
| * [[Tensor contraction]]
| |
| | |
| {{DEFAULTSORT:Interior Product}}
| |
| [[Category:Differential forms]]
| |
| [[Category:Multilinear algebra]]
| |
| | |
| | |
| {{differential-geometry-stub}}
| |
I am Celeste from Amsterdam. I am learning to play the Xylophone. Other hobbies are Knapping.
Here is my web-site - Fifa 15 Coin Generator; web38.wwwsrv03.t3-Network.de,