Interior product

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 : \Omega^p(M) \to \Omega^{p-1}(M) is the map which sends a p-form \omega to the (p - 1)-form \iota_X \omega defined by the property that (\iota_X\omega)\left(X_1, \ldots, X_{p-1}\right) = \omega\left(X, X_1, \ldots, X_{p-1}\right) for any vector fields X_1, \ldots, X_{p-1}. When \omega is a scalar field (0-form), \iota_X \omega = 0 by convention. The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms \alpha \displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha, X \rangle, where \langle \,\cdot, \cdot\, \rangle is the duality pairing between \alpha and the vector X. Explicitly, if \alpha is a p-form and \beta is a q-form, then \iota_X(\alpha \wedge \beta) = \left(\iota_X\alpha\right) \wedge \beta + (-1)^p \alpha \wedge \left(\iota_X\beta\right). 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 (x_1, \ldots, x_n) the vector field X is given by X = f_1 \frac{\partial}{\partial x_1} + \cdots + f_n \frac{\partial}{\partial x_n} then the interior product is given by \iota_X (dx_1 \wedge \cdots \wedge dx_n) = \sum_{r=1}^{n}(-1)^{r-1}f_r dx_1 \wedge \cdots \wedge \widehat{dx_r} \wedge \cdots \wedge dx_n, where dx_1\wedge \cdots \wedge \widehat{dx_r} \wedge \cdots \wedge dx_n is the form obtained by omitting dx_r from dx_1 \wedge \cdots \wedge dx_n. By antisymmetry of forms, \iota_X \iota_Y \omega = -\iota_Y \iota_X \omega, and so \iota_X \circ \iota_X = 0. This may be compared to the exterior derivative d, which has the property d \circ d = 0. The interior product with respect to the commutator of two vector fields X, Y satisfies the identity \iota_{[X,Y]} = \left[\mathcal{L}_X, \iota_Y\right] = \left[\iota_X, \mathcal{L}_Y\right]. Proof. For any k-form \Omega, \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]}\Omegaand 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 or Cartan magic formula): \mathcal L_X\omega = d(\iota_X \omega) + \iota_X d\omega = \left\{ d, \iota_X \right\} \omega. 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 momentum map. The Cartan homotopy formula is named after Élie Cartan. {{Math proof|title=Proof by direct computation |proof= Since vector fields are locally integrable, we can always find a local coordinate system (\xi^1, \dots, \xi^n) such that the vector field X corresponds to the partial derivative with respect to the first coordinate, i.e., X = \partial_1. (See Straightening theorem for vector fields) By linearity of the interior product, exterior derivative, and Lie derivative, it suffices to prove the Cartan's magic formula for monomial k-forms. There are only two cases: Case 1: \alpha = a \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k. Direct computation yields: \begin{aligned} \iota_X \alpha &= a \, d\xi^2 \wedge \dots \wedge d\xi^k, \\ d(\iota_X \alpha) &= (\partial_1 a) \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k + \sum_{i=k+1}^n (\partial_i a) \, d\xi^i \wedge d\xi^2 \wedge \dots \wedge d\xi^k, \\ d\alpha &= \sum_{i=k+1}^n (\partial_i a) \, d\xi^i \wedge d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k, \\ \iota_X(d\alpha) &= -\sum_{i=k+1}^n (\partial_i a) \, d\xi^i \wedge d\xi^2 \wedge \dots \wedge d\xi^k, \\ L_X\alpha &= (\partial_1 a) \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k. \end{aligned} Case 2: \alpha = a \, d\xi^2 \wedge d\xi^3 \wedge \dots \wedge d\xi^{k+1} . Direct computation yields: \begin{aligned} \iota_X \alpha &= 0, \\ d\alpha &= (\partial_1 a) \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^{k+1} + \sum_{i=k+2}^n (\partial_i a) \, d\xi^i \wedge d\xi^2 \wedge \dots \wedge d\xi^{k+1}, \\ \iota_X(d\alpha) &= (\partial_1 a) \, d\xi^2 \wedge \dots \wedge d\xi^{k+1}, \\ L_X\alpha &= (\partial_1 a) \, d\xi^2 \wedge \dots \wedge d\xi^{k+1}. \end{aligned} }} ==In Exterior Algebra==

In the exterior algebra over a vector space V, the interior product is generalized for arbitrary multivectors a and b. The right interior product, or right contraction, \textstyle \mathbin{\lfloor} : \bigwedge V \times \bigwedge V \to \bigwedge V is defined as : a \mathbin{\lfloor} b = a \vee b^\bigstar, where \vee is the exterior antiproduct (also known as the regressive product), and the superscript \bigstar denotes the Hodge dual. Similarly, the left interior product, or left contraction, \rfloor is defined as : a \mathbin{\rfloor} b = a_\bigstar \vee b, where the subscript \bigstar denotes the left version of the Hodge dual. When a and b are homogeneous multivectors with the same grade, then the left and right interior products each reduce to the inner product such that : a \mathbin{\lfloor} b = a \mathbin{\rfloor} b = a \cdot b. For a vector X (which has grade 1), a homogeneous multivector a having grade p, and an arbitrary multivector b, the right interior product satisfies the rule : (a \wedge b) \mathbin{\lfloor} X = (a \mathbin{\lfloor} X) \wedge b + (-1)^p a \wedge (b \mathbin{\lfloor} X). This is the exact analog of the Leibniz product rule given for the operator \iota_X above. ==See also==