Exterior calculus identities

This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (September 2023) (Learn how and when to remove this template message)

This article summarizes several identities in exterior calculus.^{[1][2][3][4][5]}

Notation

The following summarizes short definitions and notations that are used in this article.

Manifold

[math]\displaystyle{ M }[/math], [math]\displaystyle{ N }[/math] are [math]\displaystyle{ n }[/math]-dimensional smooth manifolds, where [math]\displaystyle{ n\in \mathbb{N} }[/math]. That is, differentiable manifolds that can be differentiated enough times for the purposes on this page.

[math]\displaystyle{ p \in M }[/math], [math]\displaystyle{ q \in N }[/math] denote one point on each of the manifolds.

The boundary of a manifold [math]\displaystyle{ M }[/math] is a manifold [math]\displaystyle{ \partial M }[/math], which has dimension [math]\displaystyle{ n - 1 }[/math]. An orientation on [math]\displaystyle{ M }[/math] induces an orientation on [math]\displaystyle{ \partial M }[/math].

We usually denote a submanifold by [math]\displaystyle{ \Sigma \subset M }[/math].

Tangent and cotangent bundles

[math]\displaystyle{ TM }[/math], [math]\displaystyle{ T^{*}M }[/math] denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold [math]\displaystyle{ M }[/math].

[math]\displaystyle{ T_p M }[/math], [math]\displaystyle{ T_q N }[/math] denote the tangent spaces of [math]\displaystyle{ M }[/math], [math]\displaystyle{ N }[/math] at the points [math]\displaystyle{ p }[/math], [math]\displaystyle{ q }[/math], respectively. [math]\displaystyle{ T^{*}_p M }[/math] denotes the cotangent space of [math]\displaystyle{ M }[/math] at the point [math]\displaystyle{ p }[/math].

Sections of the tangent bundles, also known as vector fields, are typically denoted as [math]\displaystyle{ X, Y, Z \in \Gamma(TM) }[/math] such that at a point [math]\displaystyle{ p \in M }[/math] we have [math]\displaystyle{ X|_p, Y|_p, Z|_p \in T_p M }[/math]. Sections of the cotangent bundle, also known as differential 1-forms (or covector fields), are typically denoted as [math]\displaystyle{ \alpha, \beta \in \Gamma(T^{*}M) }[/math] such that at a point [math]\displaystyle{ p \in M }[/math] we have [math]\displaystyle{ \alpha|_p, \beta|_p \in T^{*}_p M }[/math]. An alternative notation for [math]\displaystyle{ \Gamma(T^{*}M) }[/math] is [math]\displaystyle{ \Omega^1(M) }[/math].

Differential k-forms

Differential [math]\displaystyle{ k }[/math]-forms, which we refer to simply as [math]\displaystyle{ k }[/math]-forms here, are differential forms defined on [math]\displaystyle{ TM }[/math]. We denote the set of all [math]\displaystyle{ k }[/math]-forms as [math]\displaystyle{ \Omega^k(M) }[/math]. For [math]\displaystyle{ 0\leq k,\ l,\ m\leq n }[/math] we usually write [math]\displaystyle{ \alpha\in\Omega^k(M) }[/math], [math]\displaystyle{ \beta\in\Omega^l(M) }[/math], [math]\displaystyle{ \gamma\in\Omega^m(M) }[/math].

[math]\displaystyle{ 0 }[/math]-forms [math]\displaystyle{ f\in\Omega^0(M) }[/math] are just scalar functions [math]\displaystyle{ C^{\infty}(M) }[/math] on [math]\displaystyle{ M }[/math]. [math]\displaystyle{ \mathbf{1}\in\Omega^0(M) }[/math] denotes the constant [math]\displaystyle{ 0 }[/math]-form equal to [math]\displaystyle{ 1 }[/math] everywhere.

Omitted elements of a sequence

When we are given [math]\displaystyle{ (k+1) }[/math] inputs [math]\displaystyle{ X_0,\ldots,X_k }[/math] and a [math]\displaystyle{ k }[/math]-form [math]\displaystyle{ \alpha\in\Omega^k(M) }[/math] we denote omission of the [math]\displaystyle{ i }[/math]th entry by writing

[math]\displaystyle{ \alpha(X_0,\ldots,\hat{X}_i,\ldots,X_k):=\alpha(X_0,\ldots,X_{i-1},X_{i+1},\ldots,X_k) . }[/math]

Exterior product

The exterior product is also known as the wedge product. It is denoted by [math]\displaystyle{ \wedge : \Omega^k(M) \times \Omega^l(M) \rightarrow \Omega^{k+l}(M) }[/math]. The exterior product of a [math]\displaystyle{ k }[/math]-form [math]\displaystyle{ \alpha\in\Omega^k(M) }[/math] and an [math]\displaystyle{ l }[/math]-form [math]\displaystyle{ \beta\in\Omega^l(M) }[/math] produce a [math]\displaystyle{ (k+l) }[/math]-form [math]\displaystyle{ \alpha\wedge\beta \in\Omega^{k+l}(M) }[/math]. It can be written using the set [math]\displaystyle{ S(k,k+l) }[/math] of all permutations [math]\displaystyle{ \sigma }[/math] of [math]\displaystyle{ \{1,\ldots,n\} }[/math] such that [math]\displaystyle{ \sigma(1)\lt \ldots \lt \sigma(k), \ \sigma(k+1)\lt \ldots \lt \sigma(k+l) }[/math] as

[math]\displaystyle{ (\alpha\wedge\beta)(X_1,\ldots,X_{k+l})=\sum_{\sigma\in S(k,k+l)}\text{sign}(\sigma)\alpha(X_{\sigma(1)},\ldots,X_{\sigma(k)})\otimes\beta(X_{\sigma(k+1)},\ldots,X_{\sigma(k+l)}) . }[/math]

Directional derivative

The directional derivative of a 0-form [math]\displaystyle{ f\in\Omega^0(M) }[/math] along a section [math]\displaystyle{ X\in\Gamma(TM) }[/math] is a 0-form denoted [math]\displaystyle{ \partial_X f . }[/math]

Exterior derivative

The exterior derivative [math]\displaystyle{ d_k : \Omega^k(M) \rightarrow \Omega^{k+1}(M) }[/math] is defined for all [math]\displaystyle{ 0 \leq k\leq n }[/math]. We generally omit the subscript when it is clear from the context.

For a [math]\displaystyle{ 0 }[/math]-form [math]\displaystyle{ f\in\Omega^0(M) }[/math] we have [math]\displaystyle{ d_0f\in\Omega^1(M) }[/math] as the [math]\displaystyle{ 1 }[/math]-form that gives the directional derivative, i.e., for the section [math]\displaystyle{ X\in \Gamma(TM) }[/math] we have [math]\displaystyle{ (d_0f)(X) = \partial_X f }[/math], the directional derivative of [math]\displaystyle{ f }[/math] along [math]\displaystyle{ X }[/math].^[6]

For [math]\displaystyle{ 0 \lt k\leq n }[/math],^[6]

[math]\displaystyle{ (d_k\omega)(X_0,\ldots,X_k)=\sum_{0\leq j\leq k}(-1)^jd_{0}(\omega(X_0,\ldots,\hat{X}_j,\ldots,X_k))(X_j) + \sum_{0\leq i \lt j\leq k}(-1)^{i+j}\omega([X_i,X_j],X_0,\ldots,\hat{X}_i,\ldots,\hat{X}_j,\ldots,X_k) . }[/math]

Lie bracket

The Lie bracket of sections [math]\displaystyle{ X,Y \in \Gamma(TM) }[/math] is defined as the unique section [math]\displaystyle{ [X,Y] \in \Gamma(TM) }[/math] that satisfies

[math]\displaystyle{ \forall f\in\Omega^0(M) \Rightarrow \partial_{[X,Y]}f = \partial_X \partial_Y f - \partial_Y \partial_X f . }[/math]

Tangent maps

If [math]\displaystyle{ \phi : M \rightarrow N }[/math] is a smooth map, then [math]\displaystyle{ d\phi|_p:T_pM\rightarrow T_{\phi(p)}N }[/math] defines a tangent map from [math]\displaystyle{ M }[/math] to [math]\displaystyle{ N }[/math]. It is defined through curves [math]\displaystyle{ \gamma }[/math] on [math]\displaystyle{ M }[/math] with derivative [math]\displaystyle{ \gamma'(0)=X\in T_pM }[/math] such that

[math]\displaystyle{ d\phi(X):=(\phi\circ\gamma)' . }[/math]

Note that [math]\displaystyle{ \phi }[/math] is a [math]\displaystyle{ 0 }[/math]-form with values in [math]\displaystyle{ N }[/math].

Pull-back

If [math]\displaystyle{ \phi : M \rightarrow N }[/math] is a smooth map, then the pull-back of a [math]\displaystyle{ k }[/math]-form [math]\displaystyle{ \alpha\in \Omega^k(N) }[/math] is defined such that for any [math]\displaystyle{ k }[/math]-dimensional submanifold [math]\displaystyle{ \Sigma\subset M }[/math]

[math]\displaystyle{ \int_{\Sigma} \phi^*\alpha = \int_{\phi(\Sigma)} \alpha . }[/math]

The pull-back can also be expressed as

[math]\displaystyle{ (\phi^*\alpha)(X_1,\ldots,X_k)=\alpha(d\phi(X_1),\ldots,d\phi(X_k)) . }[/math]

Interior product

Also known as the interior derivative, the interior product given a section [math]\displaystyle{ Y\in \Gamma(TM) }[/math] is a map [math]\displaystyle{ \iota_Y:\Omega^{k+1}(M) \rightarrow \Omega^k(M) }[/math] that effectively substitutes the first input of a [math]\displaystyle{ (k+1) }[/math]-form with [math]\displaystyle{ Y }[/math]. If [math]\displaystyle{ \alpha\in\Omega^{k+1}(M) }[/math] and [math]\displaystyle{ X_i\in \Gamma(TM) }[/math] then

[math]\displaystyle{ (\iota_Y\alpha)(X_1,\ldots,X_k) = \alpha(Y,X_1,\ldots,X_k) . }[/math]

Metric tensor

Given a nondegenerate bilinear form [math]\displaystyle{ g_p( \cdot , \cdot ) }[/math] on each [math]\displaystyle{ T_p M }[/math] that is continuous on [math]\displaystyle{ M }[/math], the manifold becomes a pseudo-Riemannian manifold. We denote the metric tensor [math]\displaystyle{ g }[/math], defined pointwise by [math]\displaystyle{ g( X , Y )|_p = g_p( X|_p , Y|_p ) }[/math]. We call [math]\displaystyle{ s=\operatorname{sign}(g) }[/math] the signature of the metric. A Riemannian manifold has [math]\displaystyle{ s=1 }[/math], whereas Minkowski space has [math]\displaystyle{ s=-1 }[/math].

Musical isomorphisms

The metric tensor [math]\displaystyle{ g(\cdot,\cdot) }[/math] induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat [math]\displaystyle{ \flat }[/math] and sharp [math]\displaystyle{ \sharp }[/math]. A section [math]\displaystyle{ A \in \Gamma(TM) }[/math] corresponds to the unique one-form [math]\displaystyle{ A^{\flat}\in\Omega^1(M) }[/math] such that for all sections [math]\displaystyle{ X \in \Gamma(TM) }[/math], we have:

[math]\displaystyle{ A^{\flat}(X) = g(A,X) . }[/math]

A one-form [math]\displaystyle{ \alpha\in\Omega^1(M) }[/math] corresponds to the unique vector field [math]\displaystyle{ \alpha^{\sharp}\in \Gamma(TM) }[/math] such that for all [math]\displaystyle{ X \in \Gamma(TM) }[/math], we have:

[math]\displaystyle{ \alpha(X) = g(\alpha^\sharp,X) . }[/math]

These mappings extend via multilinearity to mappings from [math]\displaystyle{ k }[/math]-vector fields to [math]\displaystyle{ k }[/math]-forms and [math]\displaystyle{ k }[/math]-forms to [math]\displaystyle{ k }[/math]-vector fields through

[math]\displaystyle{ (A_1 \wedge A_2 \wedge \cdots \wedge A_k)^{\flat} = A_1^{\flat} \wedge A_2^{\flat} \wedge \cdots \wedge A_k^{\flat} }[/math]

[math]\displaystyle{ (\alpha_1 \wedge \alpha_2 \wedge \cdots \wedge \alpha_k)^{\sharp} = \alpha_1^{\sharp} \wedge \alpha_2^{\sharp} \wedge \cdots \wedge \alpha_k^{\sharp}. }[/math]

Hodge star

For an n-manifold M, the Hodge star operator [math]\displaystyle{ {\star}:\Omega^k(M)\rightarrow\Omega^{n-k}(M) }[/math] is a duality mapping taking a [math]\displaystyle{ k }[/math]-form [math]\displaystyle{ \alpha \in \Omega^k(M) }[/math] to an [math]\displaystyle{ (n{-}k) }[/math]-form [math]\displaystyle{ ({\star}\alpha) \in \Omega^{n-k}(M) }[/math].

It can be defined in terms of an oriented frame [math]\displaystyle{ (X_1,\ldots,X_n) }[/math] for [math]\displaystyle{ TM }[/math], orthonormal with respect to the given metric tensor [math]\displaystyle{ g }[/math]:

[math]\displaystyle{ ({\star}\alpha)(X_1,\ldots,X_{n-k})=\alpha(X_{n-k+1},\ldots,X_n) . }[/math]

Co-differential operator

The co-differential operator [math]\displaystyle{ \delta:\Omega^k(M)\rightarrow\Omega^{k-1}(M) }[/math] on an [math]\displaystyle{ n }[/math] dimensional manifold [math]\displaystyle{ M }[/math] is defined by

[math]\displaystyle{ \delta := (-1)^{k} {\star}^{-1} d {\star} = (-1)^{nk+n+1}{\star} d {\star} . }[/math]

The Hodge–Dirac operator, [math]\displaystyle{ d+\delta }[/math], is a Dirac operator studied in Clifford analysis.

Oriented manifold

An [math]\displaystyle{ n }[/math]-dimensional orientable manifold M is a manifold that can be equipped with a choice of an n-form [math]\displaystyle{ \mu\in\Omega^n(M) }[/math] that is continuous and nonzero everywhere on M.

Volume form

On an orientable manifold [math]\displaystyle{ M }[/math] the canonical choice of a volume form given a metric tensor [math]\displaystyle{ g }[/math] and an orientation is [math]\displaystyle{ \mathbf{det}:=\sqrt{|\det g|}\;dX_1^{\flat}\wedge\ldots\wedge dX_n^{\flat} }[/math] for any basis [math]\displaystyle{ dX_1,\ldots, dX_n }[/math] ordered to match the orientation.

Area form

Given a volume form [math]\displaystyle{ \mathbf{det} }[/math] and a unit normal vector [math]\displaystyle{ N }[/math] we can also define an area form [math]\displaystyle{ \sigma:=\iota_N\textbf{det} }[/math] on the boundary [math]\displaystyle{ \partial M. }[/math]

Bilinear form on k-forms

A generalization of the metric tensor, the symmetric bilinear form between two [math]\displaystyle{ k }[/math]-forms [math]\displaystyle{ \alpha,\beta\in\Omega^k(M) }[/math], is defined pointwise on [math]\displaystyle{ M }[/math] by

[math]\displaystyle{ \langle\alpha,\beta\rangle|_p := {\star}(\alpha\wedge {\star}\beta )|_p . }[/math]

The [math]\displaystyle{ L^2 }[/math]-bilinear form for the space of [math]\displaystyle{ k }[/math]-forms [math]\displaystyle{ \Omega^k(M) }[/math] is defined by

[math]\displaystyle{ \langle\!\langle\alpha,\beta\rangle\!\rangle:= \int_M\alpha\wedge {\star}\beta . }[/math]

In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).

Lie derivative

We define the Lie derivative [math]\displaystyle{ \mathcal{L}:\Omega^k(M)\rightarrow\Omega^k(M) }[/math] through Cartan's magic formula for a given section [math]\displaystyle{ X\in \Gamma(TM) }[/math] as

[math]\displaystyle{ \mathcal{L}_X = d \circ \iota_X + \iota_X \circ d . }[/math]

It describes the change of a [math]\displaystyle{ k }[/math]-form along a flow [math]\displaystyle{ \phi_t }[/math] associated to the section [math]\displaystyle{ X }[/math].

Laplace–Beltrami operator

The Laplacian [math]\displaystyle{ \Delta:\Omega^k(M) \rightarrow \Omega^k(M) }[/math] is defined as [math]\displaystyle{ \Delta = -(d\delta + \delta d) }[/math].

Important definitions

Definitions on Ω^k(M)

[math]\displaystyle{ \alpha\in\Omega^k(M) }[/math] is called...

• closed if [math]\displaystyle{ d\alpha=0 }[/math]
• exact if [math]\displaystyle{ \alpha = d\beta }[/math] for some [math]\displaystyle{ \beta\in\Omega^{k-1} }[/math]
• coclosed if [math]\displaystyle{ \delta\alpha=0 }[/math]
• coexact if [math]\displaystyle{ \alpha = \delta\beta }[/math] for some [math]\displaystyle{ \beta\in\Omega^{k+1} }[/math]
• harmonic if closed and coclosed

Cohomology

The [math]\displaystyle{ k }[/math]-th cohomology of a manifold [math]\displaystyle{ M }[/math] and its exterior derivative operators [math]\displaystyle{ d_0,\ldots,d_{n-1} }[/math] is given by

[math]\displaystyle{ H^k(M):=\frac{\text{ker}(d_{k})}{\text{im}(d_{k-1})} }[/math]

Two closed [math]\displaystyle{ k }[/math]-forms [math]\displaystyle{ \alpha,\beta\in\Omega^k(M) }[/math] are in the same cohomology class if their difference is an exact form i.e.

[math]\displaystyle{ [\alpha]=[\beta] \ \ \Longleftrightarrow\ \ \alpha{-}\beta = d\eta \ \text{ for some } \eta\in\Omega^{k-1}(M) }[/math]

A closed surface of genus [math]\displaystyle{ g }[/math] will have [math]\displaystyle{ 2g }[/math] generators which are harmonic.

Dirichlet energy

Given [math]\displaystyle{ \alpha\in\Omega^k(M) }[/math], its Dirichlet energy is

[math]\displaystyle{ \mathcal{E}_\text{D}(\alpha):= \dfrac{1}{2}\langle\!\langle d\alpha,d\alpha\rangle\!\rangle + \dfrac{1}{2}\langle\!\langle \delta\alpha,\delta\alpha\rangle\!\rangle }[/math]

Properties

Exterior derivative properties

[math]\displaystyle{ \int_{\Sigma} d\alpha = \int_{\partial\Sigma} \alpha }[/math] ( Stokes' theorem )

[math]\displaystyle{ d \circ d = 0 }[/math] ( cochain complex )

[math]\displaystyle{ d(\alpha \wedge \beta ) = d\alpha\wedge \beta +(-1)^k\alpha\wedge d\beta }[/math] for [math]\displaystyle{ \alpha\in\Omega^k(M), \ \beta\in\Omega^l(M) }[/math] ( Leibniz rule )

[math]\displaystyle{ df(X) = \partial_X f }[/math] for [math]\displaystyle{ f\in\Omega^0(M), \ X\in \Gamma(TM) }[/math] ( directional derivative )

[math]\displaystyle{ d\alpha = 0 }[/math] for [math]\displaystyle{ \alpha \in \Omega^n(M), \ \text{dim}(M)=n }[/math]

Exterior product properties

[math]\displaystyle{ \alpha \wedge \beta = (-1)^{kl}\beta \wedge \alpha }[/math] for [math]\displaystyle{ \alpha\in\Omega^k(M), \ \beta\in\Omega^l(M) }[/math] ( alternating )

[math]\displaystyle{ (\alpha \wedge \beta)\wedge\gamma = \alpha \wedge (\beta\wedge\gamma) }[/math] ( associativity )

[math]\displaystyle{ (\lambda\alpha) \wedge \beta = \lambda (\alpha \wedge \beta) }[/math] for [math]\displaystyle{ \lambda\in\mathbb{R} }[/math] ( compatibility of scalar multiplication )

[math]\displaystyle{ \alpha \wedge ( \beta_1 + \beta_2 ) = \alpha \wedge \beta_1 + \alpha \wedge \beta_2 }[/math] ( distributivity over addition )

[math]\displaystyle{ \alpha \wedge \alpha = 0 }[/math] for [math]\displaystyle{ \alpha\in\Omega^k(M) }[/math] when [math]\displaystyle{ k }[/math] is odd or [math]\displaystyle{ \operatorname{rank} \alpha \le 1 }[/math]. The rank of a [math]\displaystyle{ k }[/math]-form [math]\displaystyle{ \alpha }[/math] means the minimum number of monomial terms (exterior products of one-forms) that must be summed to produce [math]\displaystyle{ \alpha }[/math].

Pull-back properties

[math]\displaystyle{ d(\phi^*\alpha) = \phi^*(d\alpha) }[/math] ( commutative with [math]\displaystyle{ d }[/math] )

[math]\displaystyle{ \phi^*(\alpha\wedge\beta) = (\phi^*\alpha)\wedge(\phi^*\beta) }[/math] ( distributes over [math]\displaystyle{ \wedge }[/math] )

[math]\displaystyle{ (\phi_1\circ\phi_2)^* = \phi_2^*\phi_1^* }[/math] ( contravariant )

[math]\displaystyle{ \phi^*f=f\circ\phi }[/math] for [math]\displaystyle{ f\in\Omega^0(N) }[/math] ( function composition )

Musical isomorphism properties

[math]\displaystyle{ (X^{\flat})^{\sharp}=X }[/math]

[math]\displaystyle{ (\alpha^{\sharp})^{\flat}=\alpha }[/math]

Interior product properties

[math]\displaystyle{ \iota_X \circ \iota_X = 0 }[/math] ( nilpotent )

[math]\displaystyle{ \iota_X \circ \iota_Y = - \iota_Y \circ \iota_X }[/math]

[math]\displaystyle{ \iota_X (\alpha \wedge \beta ) = (\iota_X\alpha)\wedge\beta + (-1)^k\alpha\wedge(\iota_X \beta ) }[/math] for [math]\displaystyle{ \alpha\in\Omega^k(M), \ \beta\in\Omega^l(M) }[/math] ( Leibniz rule )

[math]\displaystyle{ \iota_X\alpha = \alpha(X) }[/math] for [math]\displaystyle{ \alpha\in\Omega^1(M) }[/math]

[math]\displaystyle{ \iota_X f = 0 }[/math] for [math]\displaystyle{ f \in \Omega^0(M) }[/math]

[math]\displaystyle{ \iota_X(f\alpha) = f \iota_X\alpha }[/math] for [math]\displaystyle{ f \in \Omega^0(M) }[/math]

Hodge star properties

[math]\displaystyle{ {\star}(\lambda_1\alpha + \lambda_2\beta) = \lambda_1({\star}\alpha) + \lambda_2({\star}\beta) }[/math] for [math]\displaystyle{ \lambda_1,\lambda_2\in\mathbb{R} }[/math] ( linearity )

[math]\displaystyle{ {\star}{\star}\alpha = s(-1)^{k(n-k)}\alpha }[/math] for [math]\displaystyle{ \alpha\in \Omega^k(M) }[/math], [math]\displaystyle{ n=\dim(M) }[/math], and [math]\displaystyle{ s = \operatorname{sign}(g) }[/math] the sign of the metric

[math]\displaystyle{ {\star}^{(-1)} = s(-1)^{k(n-k)}{\star} }[/math] ( inversion )

[math]\displaystyle{ {\star}(f\alpha)=f({\star}\alpha) }[/math] for [math]\displaystyle{ f\in\Omega^0(M) }[/math] ( commutative with [math]\displaystyle{ 0 }[/math]-forms )

[math]\displaystyle{ \langle\!\langle\alpha,\alpha\rangle\!\rangle = \langle\!\langle{\star}\alpha,{\star}\alpha\rangle\!\rangle }[/math] for [math]\displaystyle{ \alpha\in\Omega^1(M) }[/math] ( Hodge star preserves [math]\displaystyle{ 1 }[/math]-form norm )

[math]\displaystyle{ {\star} \mathbf{1} = \mathbf{det} }[/math] ( Hodge dual of constant function 1 is the volume form )

Co-differential operator properties

[math]\displaystyle{ \delta\circ\delta = 0 }[/math] ( nilpotent )

[math]\displaystyle{ {\star}\delta=(-1)^kd{\star} }[/math] and [math]\displaystyle{ {\star} d = (-1)^{k+1}\delta{\star} }[/math] ( Hodge adjoint to [math]\displaystyle{ d }[/math] )

[math]\displaystyle{ \langle\!\langle d\alpha,\beta\rangle\!\rangle = \langle\!\langle \alpha,\delta\beta\rangle\!\rangle }[/math] if [math]\displaystyle{ \partial M=0 }[/math] ( [math]\displaystyle{ \delta }[/math] adjoint to [math]\displaystyle{ d }[/math] )

In general, [math]\displaystyle{ \int_M d\alpha \wedge \star \beta = \int_{\partial M} \alpha \wedge \star \beta + \int_M \alpha\wedge\star\delta\beta }[/math]

[math]\displaystyle{ \delta f = 0 }[/math] for [math]\displaystyle{ f \in \Omega^0(M) }[/math]

Lie derivative properties

[math]\displaystyle{ d\circ\mathcal{L}_X = \mathcal{L}_X\circ d }[/math] ( commutative with [math]\displaystyle{ d }[/math] )

[math]\displaystyle{ \iota_X \circ\mathcal{L}_X = \mathcal{L}_X\circ \iota_X }[/math] ( commutative with [math]\displaystyle{ \iota_X }[/math] )

[math]\displaystyle{ \mathcal{L}_X(\iota_Y\alpha) = \iota_{[X,Y]}\alpha + \iota_Y\mathcal{L}_X\alpha }[/math]

[math]\displaystyle{ \mathcal{L}_X(\alpha\wedge\beta) = (\mathcal{L}_X\alpha)\wedge\beta + \alpha\wedge(\mathcal{L}_X\beta) }[/math] ( Leibniz rule )

Exterior calculus identities

[math]\displaystyle{ \iota_X({\star}\mathbf{1}) = {\star} X^{\flat} }[/math]

[math]\displaystyle{ \iota_X({\star}\alpha) = (-1)^k{\star}(X^{\flat}\wedge\alpha) }[/math] if [math]\displaystyle{ \alpha\in\Omega^k(M) }[/math]

[math]\displaystyle{ \iota_X(\phi^*\alpha)=\phi^*(\iota_{d\phi(X)}\alpha) }[/math]

[math]\displaystyle{ \nu,\mu\in\Omega^n(M), \mu \text{ non-zero } \ \Rightarrow \ \exist \ f\in\Omega^0(M): \ \nu=f\mu }[/math]

[math]\displaystyle{ X^{\flat}\wedge{\star} Y^{\flat} = g(X,Y)( {\star} \mathbf{1}) }[/math] ( bilinear form )

[math]\displaystyle{ [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]] = 0 }[/math] ( Jacobi identity )

Dimensions

If [math]\displaystyle{ n=\dim M }[/math]

[math]\displaystyle{ \dim\Omega^k(M) = \binom{n}{k} }[/math] for [math]\displaystyle{ 0\leq k\leq n }[/math]

[math]\displaystyle{ \dim\Omega^k(M) = 0 }[/math] for [math]\displaystyle{ k \lt 0, \ k \gt n }[/math]

If [math]\displaystyle{ X_1,\ldots,X_n\in \Gamma(TM) }[/math] is a basis, then a basis of [math]\displaystyle{ \Omega^k(M) }[/math] is

[math]\displaystyle{ \{X_{\sigma(1)}^{\flat}\wedge\ldots\wedge X_{\sigma(k)}^{\flat} \ : \ \sigma\in S(k,n)\} }[/math]

Exterior products

Let [math]\displaystyle{ \alpha, \beta, \gamma,\alpha_i\in \Omega^1(M) }[/math] and [math]\displaystyle{ X,Y,Z,X_i }[/math] be vector fields.

[math]\displaystyle{ \alpha(X) = \det \begin{bmatrix} \alpha(X) \\ \end{bmatrix} }[/math]

[math]\displaystyle{ (\alpha\wedge\beta)(X,Y) = \det \begin{bmatrix} \alpha(X) & \alpha(Y) \\ \beta(X) & \beta(Y) \\ \end{bmatrix} }[/math]

[math]\displaystyle{ (\alpha\wedge\beta\wedge\gamma)(X,Y,Z) = \det \begin{bmatrix} \alpha(X) & \alpha(Y) & \alpha(Z) \\ \beta(X) & \beta(Y) & \beta(Z) \\ \gamma(X) & \gamma(Y) & \gamma(Z) \end{bmatrix} }[/math]

[math]\displaystyle{ (\alpha_1\wedge\ldots\wedge\alpha_l)(X_1,\ldots,X_l) = \det \begin{bmatrix} \alpha_1(X_1) & \alpha_1(X_2) & \dots & \alpha_1(X_l) \\ \alpha_2(X_1) & \alpha_2(X_2) & \dots & \alpha_2(X_l) \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_l(X_1) & \alpha_l(X_2) & \dots & \alpha_l(X_l) \end{bmatrix} }[/math]

Projection and rejection

[math]\displaystyle{ (-1)^k\iota_X{\star}\alpha = {\star}(X^{\flat}\wedge\alpha) }[/math] ( interior product [math]\displaystyle{ \iota_X{\star} }[/math] dual to wedge [math]\displaystyle{ X^{\flat}\wedge }[/math] )

[math]\displaystyle{ (\iota_X\alpha)\wedge{\star}\beta =\alpha\wedge{\star}(X^{\flat}\wedge\beta) }[/math] for [math]\displaystyle{ \alpha\in\Omega^{k+1}(M),\beta\in\Omega^k(M) }[/math]

If [math]\displaystyle{ |X|=1, \ \alpha\in\Omega^k(M) }[/math], then

• [math]\displaystyle{ \iota_X\circ (X^{\flat}\wedge ):\Omega^k(M)\rightarrow\Omega^k(M) }[/math] is the projection of [math]\displaystyle{ \alpha }[/math] onto the orthogonal complement of [math]\displaystyle{ X }[/math].
• [math]\displaystyle{ (X^{\flat}\wedge )\circ \iota_X:\Omega^k(M)\rightarrow\Omega^k(M) }[/math] is the rejection of [math]\displaystyle{ \alpha }[/math], the remainder of the projection.
• thus [math]\displaystyle{ \iota_X \circ (X^{\flat}\wedge ) + (X^{\flat}\wedge)\circ\iota_X = \text{id} }[/math] ( projection–rejection decomposition )

Given the boundary [math]\displaystyle{ \partial M }[/math] with unit normal vector [math]\displaystyle{ N }[/math]

• [math]\displaystyle{ \mathbf{t}:=\iota_N\circ (N^{\flat}\wedge ) }[/math] extracts the tangential component of the boundary.
• [math]\displaystyle{ \mathbf{n}:=(\text{id}-\mathbf{t}) }[/math] extracts the normal component of the boundary.

Sum expressions

[math]\displaystyle{ (d\alpha)(X_0,\ldots,X_k)=\sum_{0\leq j\leq k}(-1)^jd(\alpha(X_0,\ldots,\hat{X}_j,\ldots,X_k))(X_j) + \sum_{0\leq i \lt j\leq k}(-1)^{i+j}\alpha([X_i,X_j],X_0,\ldots,\hat{X}_i,\ldots,\hat{X}_j,\ldots,X_k) }[/math]

[math]\displaystyle{ (d\alpha)(X_1,\ldots,X_k) =\sum_{i=1}^k(-1)^{i+1}(\nabla_{X_i}\alpha)(X_1,\ldots,\hat{X}_i,\ldots,X_k) }[/math]

[math]\displaystyle{ (\delta\alpha)(X_1,\ldots,X_{k-1})=-\sum_{i=1}^n(\iota_{E_i}(\nabla_{E_i}\alpha))(X_1,\ldots,\hat{X}_i,\ldots,X_k) }[/math] given a positively oriented orthonormal frame [math]\displaystyle{ E_1,\ldots,E_n }[/math].

[math]\displaystyle{ (\mathcal{L}_Y\alpha)(X_1,\ldots,X_k) =(\nabla_Y\alpha)(X_1,\ldots,X_k) - \sum_{i=1}^k\alpha(X_1,\ldots,\nabla_{X_i}Y,\ldots,X_k) }[/math]

Hodge decomposition

If [math]\displaystyle{ \partial M =\empty }[/math], [math]\displaystyle{ \omega\in\Omega^k(M) \Rightarrow \exists \alpha\in\Omega^{k-1}, \ \beta\in\Omega^{k+1}, \ \gamma\in\Omega^k(M), \ d\gamma=0, \ \delta\gamma = 0 }[/math] such that^{[citation needed]}

[math]\displaystyle{ \omega = d\alpha + \delta\beta + \gamma }[/math]

Poincaré lemma

If a boundaryless manifold [math]\displaystyle{ M }[/math] has trivial cohomology [math]\displaystyle{ H^k(M)=\{0\} }[/math], then any closed [math]\displaystyle{ \omega\in\Omega^k(M) }[/math] is exact. This is the case if M is contractible.

Relations to vector calculus

Identities in Euclidean 3-space

Let Euclidean metric [math]\displaystyle{ g(X,Y):=\langle X,Y\rangle = X\cdot Y }[/math].

We use [math]\displaystyle{ \nabla = \left( {\partial \over \partial x}, {\partial \over \partial y}, {\partial \over \partial z} \right) }[/math] differential operator [math]\displaystyle{ \mathbb{R}^3 }[/math]

[math]\displaystyle{ \iota_X\alpha = g(X,\alpha^{\sharp}) = X\cdot \alpha^{\sharp} }[/math] for [math]\displaystyle{ \alpha\in\Omega^1(M) }[/math].

[math]\displaystyle{ \mathbf{det}(X,Y,Z)=\langle X,Y\times Z\rangle = \langle X\times Y,Z\rangle }[/math] ( scalar triple product )

[math]\displaystyle{ X\times Y = ({\star}(X^{\flat}\wedge Y^{\flat}))^{\sharp} }[/math] ( cross product )

[math]\displaystyle{ \iota_X\alpha=-(X\times A)^{\flat} }[/math] if [math]\displaystyle{ \alpha\in\Omega^2(M),\ A=({\star}\alpha)^{\sharp} }[/math]

[math]\displaystyle{ X\cdot Y = {\star}(X^{\flat}\wedge {\star} Y^{\flat}) }[/math] ( scalar product )

[math]\displaystyle{ \nabla f=(df)^{\sharp} }[/math] ( gradient )

[math]\displaystyle{ X\cdot\nabla f=df(X) }[/math] ( directional derivative )

[math]\displaystyle{ \nabla\cdot X = {\star} d {\star} X^{\flat} = -\delta X^{\flat} }[/math] ( divergence )

[math]\displaystyle{ \nabla\times X = ({\star} d X^{\flat})^{\sharp} }[/math] ( curl )

[math]\displaystyle{ \langle X,N\rangle\sigma = {\star} X^\flat }[/math] where [math]\displaystyle{ N }[/math] is the unit normal vector of [math]\displaystyle{ \partial M }[/math] and [math]\displaystyle{ \sigma=\iota_{N}\mathbf{det} }[/math] is the area form on [math]\displaystyle{ \partial M }[/math].

[math]\displaystyle{ \int_{\Sigma} d{\star} X^{\flat} = \int_{\partial\Sigma}{\star} X^{\flat} = \int_{\partial\Sigma}\langle X,N\rangle\sigma }[/math] ( divergence theorem )

Lie derivatives

[math]\displaystyle{ \mathcal{L}_X f =X\cdot \nabla f }[/math] ( [math]\displaystyle{ 0 }[/math]-forms )

[math]\displaystyle{ \mathcal{L}_X \alpha = (\nabla_X\alpha^{\sharp})^{\flat} +g(\alpha^{\sharp},\nabla X) }[/math] ( [math]\displaystyle{ 1 }[/math]-forms )

[math]\displaystyle{ {\star}\mathcal{L}_X\beta = \left( \nabla_XB - \nabla_BX + (\text{div}X)B \right)^{\flat} }[/math] if [math]\displaystyle{ B=({\star}\beta)^{\sharp} }[/math] ( [math]\displaystyle{ 2 }[/math]-forms on [math]\displaystyle{ 3 }[/math]-manifolds )

[math]\displaystyle{ {\star}\mathcal{L}_X\rho = dq(X)+(\text{div}X)q }[/math] if [math]\displaystyle{ \rho={\star} q \in \Omega^0(M) }[/math] ( [math]\displaystyle{ n }[/math]-forms )

[math]\displaystyle{ \mathcal{L}_X(\mathbf{det})=(\text{div}(X))\mathbf{det} }[/math]

References

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Exterior calculus identities. Read more