Exterior calculus identities

Short description: none

This article summarizes several identities in exterior calculus.

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].

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

[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

[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]