Exterior calculus identities

This article summarizes several identities in exterior calculus.

Notation

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

Manifold

$M$, $N$ are $n$-dimensional smooth manifolds, where $n\in \mathbb$. That is, differentiable manifolds that can be differentiated enough times for the purposes on this page.

$p \in M$, $q \in N$ denote one point on each of the manifolds.

The boundary of a manifold $M$ is a manifold $\partial M$, which has dimension $n - 1$. An orientation on $M$ induces an orientation on $\partial M$.

We usually denote a submanifold by $\Sigma \subset M$.

Tangent and cotangent bundles

$TM$, $T^M$ denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold $M$.

$T_p M$, $T_q N$ denote the tangent spaces of $M$, $N$ at the points $p$, $q$, respectively. $T^_p M$ denotes the cotangent space of $M$ at the point $p$.

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

Differential ''k''-forms

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

$0$-forms $f\in\Omega^0(M)$ are just scalar functions $C^(M)$ on $M$. $\mathbf\in\Omega^0(M)$ denotes the constant $0$-form equal to $1$ everywhere.

Omitted elements of a sequence

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

:$\alpha(X_0,\ldots,\hat_i,\ldots,X_k):=\alpha(X_0,\ldots,X_,X_,\ldots,X_k) .$

Exterior product

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

:$(\alpha\wedge\beta)(X_1,\ldots,X_)=\sum_\text(\sigma)\alpha(X_,\ldots,X_)\otimes\beta(X_,\ldots,X_) .$

Directional derivative

The directional derivative of a 0-form $f\in\Omega^0(M)$ along a section $X\in\Gamma(TM)$ is a 0-form denoted $\partial_X f .$

Exterior derivative

The exterior derivative $d_k : \Omega^k(M) \rightarrow \Omega^(M)$ is defined for all $0 \leq k\leq n$. We generally omit the subscript when it is clear from the context.

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

For $0 < k\leq n$,

: (d_k\omega)(X_0,\ldots,X_k)=\sum_(-1)^jd_(\omega(X_0,\ldots,\hat_j,\ldots,X_k))(X_j) + \sum_(-1)^\omega( _i,X_jX_0,\ldots,\hat_i,\ldots,\hat_j,\ldots,X_k) .

Lie bracket

The Lie bracket of sections $X,Y \in \Gamma(TM)$ is defined as the unique section ,Y\in \Gamma(TM) that satisfies

:$\forall f\in\Omega^0(M) \Rightarrow \partial_f = \partial_X \partial_Y f - \partial_Y \partial_X f .$

Tangent maps

If $\phi : M \rightarrow N$ is a smooth map, then $d\phi, _p:T_pM\rightarrow T_N$ defines a tangent map from $M$ to $N$. It is defined through curves $\gamma$ on $M$ with derivative $\gamma'(0)=X\in T_pM$ such that

:$d\phi(X):=(\phi\circ\gamma)' .$

Note that $\phi$ is a $0$-form with values in $N$.

Pull-back

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

:$\int_ \phi^*\alpha = \int_ \alpha .$

The pull-back can also be expressed as

:$(\phi^*\alpha)(X_1,\ldots,X_k)=\alpha(d\phi(X_1),\ldots,d\phi(X_k)) .$

Interior product

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

:$(\iota_Y\alpha)(X_1,\ldots,X_k) = \alpha(Y,X_1,\ldots,X_k) .$

Metric tensor

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

Musical isomorphisms

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

:$A^(X) = g(A,X) .$

A one-form $\alpha\in\Omega^1(M)$ corresponds to the unique vector field $\alpha^\in \Gamma(TM)$ such that for all $X \in \Gamma(TM)$, we have:

:$\alpha(X) = g(\alpha^\sharp,X) .$

These mappings extend via multilinearity to mappings from $k$-vector fields to $k$-forms and $k$-forms to $k$-vector fields through

:$(A_1 \wedge A_2 \wedge \cdots \wedge A_k)^ = A_1^ \wedge A_2^ \wedge \cdots \wedge A_k^$
:$(\alpha_1 \wedge \alpha_2 \wedge \cdots \wedge \alpha_k)^ = \alpha_1^ \wedge \alpha_2^ \wedge \cdots \wedge \alpha_k^.$

Hodge star

For an ''n''-manifold ''M'', the Hodge star operator $:\Omega^k(M)\rightarrow\Omega^(M)$ is a duality mapping taking a $k$-form $\alpha \in \Omega^k(M)$ to an $(nk)$-form $(\alpha) \in \Omega^(M)$.

It can be defined in terms of an oriented frame $(X_1,\ldots,X_n)$ for $TM$, orthonormal with respect to the given metric tensor $g$:

:$(\alpha)(X_1,\ldots,X_)=\alpha(X_,\ldots,X_n) .$

Co-differential operator

The co-differential operator $\delta:\Omega^k(M)\rightarrow\Omega^(M)$ on an $n$ dimensional manifold $M$ is defined by

:$\delta := (-1)^ ^ d = (-1)^ d .$

The Hodge–Dirac operator, $d+\delta$, is a Dirac operator studied in Clifford analysis.

Oriented manifold

An $n$-dimensional orientable manifold is a manifold that can be equipped with a choice of an -form $\mu\in\Omega^n(M)$ that is continuous and nonzero everywhere on .

Volume form

On an orientable manifold $M$ the canonical choice of a volume form given a metric tensor $g$ and an orientation is $\mathbf:=\sqrt\;dX_1^\wedge\ldots\wedge dX_n^$ for any basis $dX_1,\ldots, dX_n$ ordered to match the orientation.

Area form

Given a volume form $\mathbf$ and a unit normal vector $N$ we can also define an area form $\sigma:=\iota_N\textbf$ on the

Bilinear form on ''k''-forms

A generalization of the metric tensor, the symmetric bilinear form between two $k$-forms $\alpha,\beta\in\Omega^k(M)$, is defined pointwise on $M$ by

:$\langle\alpha,\beta\rangle, _p := (\alpha\wedge \beta ), _p .$

The $L^2$-bilinear form for the space of $k$-forms $\Omega^k(M)$ is defined by

:$\langle\!\langle\alpha,\beta\rangle\!\rangle:= \int_M\alpha\wedge \beta .$

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

Lie derivative

We define the Lie derivative $\mathcal:\Omega^k(M)\rightarrow\Omega^k(M)$ through Cartan's magic formula for a given section $X\in \Gamma(TM)$ as

:$\mathcal_X = d \circ \iota_X + \iota_X \circ d .$

It describes the change of a $k$-form along a flow $\phi_t$ associated to the section $X$.

Laplace–Beltrami operator

The Laplacian $\Delta:\Omega^k(M) \rightarrow \Omega^k(M)$ is defined as $\Delta = -(d\delta + \delta d)$.

Important definitions

Definitions on Ω^''k''(''M'')

$\alpha\in\Omega^k(M)$ is called...

* ''closed'' if $d\alpha=0$
* ''exact'' if $\alpha = d\beta$ for some $\beta\in\Omega^$
* ''coclosed'' if $\delta\alpha=0$
* ''coexact'' if $\alpha = \delta\beta$ for some $\beta\in\Omega^$
* ''harmonic'' if ''closed'' and ''coclosed''

Cohomology

The $k$-th cohomology of a manifold $M$ and its exterior derivative operators $d_0,\ldots,d_$ is given by

:$H^k(M):=\frac$

Two closed $k$-forms $\alpha,\beta\in\Omega^k(M)$ are in the same cohomology class if their difference is an exact form i.e.

:
alphabeta\ \ \Longleftrightarrow\ \ \alpha\beta = d\eta \ \text \eta\in\Omega^(M)

A closed surface of genus $g$ will have $2g$ generators which are harmonic.

Dirichlet energy

Given $\alpha\in\Omega^k(M)$, its Dirichlet energy is

:$\mathcal_\text(\alpha):= \dfrac\langle\!\langle d\alpha,d\alpha\rangle\!\rangle + \dfrac\langle\!\langle \delta\alpha,\delta\alpha\rangle\!\rangle$

Properties

Exterior derivative properties

:$\int_ d\alpha = \int_ \alpha$ ( ''Stokes' theorem'' )

:$d \circ d = 0$ ( ''cochain complex'' )

:$d(\alpha \wedge \beta ) = d\alpha\wedge \beta +(-1)^k\alpha\wedge d\beta$ for $\alpha\in\Omega^k(M), \ \beta\in\Omega^l(M)$ ( ''Leibniz rule'' )

:$df(X) = \partial_X f$ for $f\in\Omega^0(M), \ X\in \Gamma(TM)$ ( ''directional derivative'' )

:$d\alpha = 0$ for $\alpha \in \Omega^n(M), \ \text(M)=n$

Exterior product properties

:$\alpha \wedge \beta = (-1)^\beta \wedge \alpha$ for $\alpha\in\Omega^k(M), \ \beta\in\Omega^l(M)$ ( ''alternating'' )

:$(\alpha \wedge \beta)\wedge\gamma = \alpha \wedge (\beta\wedge\gamma)$ ( ''associativity'' )

:$(\lambda\alpha) \wedge \beta = \lambda (\alpha \wedge \beta)$ for $\lambda\in\mathbb$ ( ''compatibility of scalar multiplication'' )

:$\alpha \wedge ( \beta_1 + \beta_2 ) = \alpha \wedge \beta_1 + \alpha \wedge \beta_2$ ( ''distributivity over addition'' )

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

Pull-back properties

:$d(\phi^*\alpha) = \phi^*(d\alpha)$ ( ''commutative with $d$'' )

:$\phi^*(\alpha\wedge\beta) = (\phi^*\alpha)\wedge(\phi^*\beta)$ ( ''distributes over $\wedge$'' )

:$(\phi_1\circ\phi_2)^* = \phi_2^*\phi_1^*$ ( ''contravariant'' )

:$\phi^*f=f\circ\phi$ for $f\in\Omega^0(N)$ ( ''function composition'' )

Musical isomorphism properties

:$(X^)^=X$

:$(\alpha^)^=\alpha$

Interior product properties

:$\iota_X \circ \iota_X = 0$ ( ''nilpotent'' )

:$\iota_X \circ \iota_Y = - \iota_Y \circ \iota_X$

:$\iota_X (\alpha \wedge \beta ) = (\iota_X\alpha)\wedge\beta + (-1)^k\alpha\wedge(\iota_X \beta )$ for $\alpha\in\Omega^k(M), \ \beta\in\Omega^l(M)$ ( ''Leibniz rule'' )

:$\iota_X\alpha = \alpha(X)$ for $\alpha\in\Omega^1(M)$

:$\iota_X f = 0$ for $f \in \Omega^0(M)$

:$\iota_X(f\alpha) = f \iota_X\alpha$ for $f \in \Omega^0(M)$

Hodge star properties

:$(\lambda_1\alpha + \lambda_2\beta) = \lambda_1(\alpha) + \lambda_2(\beta)$ for $\lambda_1,\lambda_2\in\mathbb$ ( ''linearity'' )

:$\alpha = s(-1)^\alpha$ for $\alpha\in \Omega^k(M)$, $n=\dim(M)$, and $s = \operatorname(g)$ the sign of the metric

:$^ = s(-1)^$ ( ''inversion'' )

:$(f\alpha)=f(\alpha)$ for $f\in\Omega^0(M)$ ( ''commutative with $0$-forms'' )

:$\langle\!\langle\alpha,\alpha\rangle\!\rangle = \langle\!\langle\alpha,\alpha\rangle\!\rangle$ for $\alpha\in\Omega^1(M)$ ( ''Hodge star preserves $1$-form norm'' )

:$\mathbf = \mathbf$ ( ''Hodge dual of constant function 1 is the volume form'' )

Co-differential operator properties

:$\delta\circ\delta = 0$ ( ''nilpotent'' )

:$\delta=(-1)^kd$ and $d = (-1)^\delta$ ( ''Hodge adjoint to $d$'' )

:$\langle\!\langle d\alpha,\beta\rangle\!\rangle = \langle\!\langle \alpha,\delta\beta\rangle\!\rangle$ if $\partial M=0$ ( ''$\delta$ adjoint to $d$'' )

:In general, $\int_M d\alpha \wedge \star \beta = \int_ \alpha \wedge \star \beta + \int_M \alpha\wedge\star\delta\beta$

:$\delta f = 0$ for $f \in \Omega^0(M)$

Lie derivative properties

:$d\circ\mathcal_X = \mathcal_X\circ d$ ( ''commutative with $d$'' )

:$\iota_X \circ\mathcal_X = \mathcal_X\circ \iota_X$ ( ''commutative with $\iota_X$'' )

:$\mathcal_X(\iota_Y\alpha) = \iota_\alpha + \iota_Y\mathcal_X\alpha$

:$\mathcal_X(\alpha\wedge\beta) = (\mathcal_X\alpha)\wedge\beta + \alpha\wedge(\mathcal_X\beta)$ ( ''Leibniz rule'' )

Exterior calculus identities

:$\iota_X(\mathbf) = X^$

:$\iota_X(\alpha) = (-1)^k(X^\wedge\alpha)$ if $\alpha\in\Omega^k(M)$

:$\iota_X(\phi^*\alpha)=\phi^*(\iota_\alpha)$

:$\nu,\mu\in\Omega^n(M), \mu \text \ \Rightarrow \ \exist \ f\in\Omega^0(M): \ \nu=f\mu$

:$X^\wedge Y^ = g(X,Y)( \mathbf)$ ( ''bilinear form'' )

:
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( ''Jacobi identity'' )

Dimensions

If $n=\dim M$

:$\dim\Omega^k(M) = \binom$ for $0\leq k\leq n$

:$\dim\Omega^k(M) = 0$ for $k < 0, \ k > n$

If $X_1,\ldots,X_n\in \Gamma(TM)$ is a basis, then a basis of $\Omega^k(M)$ is

:$\$

Exterior products

Let $\alpha, \beta, \gamma,\alpha_i\in \Omega^1(M)$ and $X,Y,Z,X_i$ be vector fields.

:$\alpha(X) = \det \begin \alpha(X) \\ \end$

:$(\alpha\wedge\beta)(X,Y) = \det \begin \alpha(X) & \alpha(Y) \\ \beta(X) & \beta(Y) \\ \end$

:$(\alpha\wedge\beta\wedge\gamma)(X,Y,Z) = \det \begin \alpha(X) & \alpha(Y) & \alpha(Z) \\ \beta(X) & \beta(Y) & \beta(Z) \\ \gamma(X) & \gamma(Y) & \gamma(Z) \end$

:$(\alpha_1\wedge\ldots\wedge\alpha_l)(X_1,\ldots,X_l) = \det \begin \alpha_1(X_1) & \alpha_1(X_2) & \dots & \alpha_1(X_l) \\ \alpha_2(X_1) & \alpha_2(X_2) & \dots & \alpha_2(X_l) \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_l(X_1) & \alpha_l(X_2) & \dots & \alpha_l(X_l) \end$

Projection and rejection

:$(-1)^k\iota_X\alpha = (X^\wedge\alpha)$ ( ''interior product $\iota_X$ dual to wedge $X^\wedge$'' )

:$(\iota_X\alpha)\wedge\beta =\alpha\wedge(X^\wedge\beta)$ for $\alpha\in\Omega^(M),\beta\in\Omega^k(M)$

If $, X, =1, \ \alpha\in\Omega^k(M)$, then

*$\iota_X\circ (X^\wedge ):\Omega^k(M)\rightarrow\Omega^k(M)$ is the ''projection'' of $\alpha$ onto the orthogonal complement of $X$.
*$(X^\wedge )\circ \iota_X:\Omega^k(M)\rightarrow\Omega^k(M)$ is the ''rejection'' of $\alpha$, the remainder of the projection.
* thus $\iota_X \circ (X^\wedge ) + (X^\wedge)\circ\iota_X = \text$ ( ''projection–rejection decomposition'' )

Given the boundary $\partial M$ with unit normal vector $N$

*$\mathbf:=\iota_N\circ (N^\wedge )$ extracts the ''tangential component'' of the boundary.
*$\mathbf:=(\text-\mathbf)$ extracts the ''normal component'' of the boundary.

Sum expressions

:
(d\alpha)(X_0,\ldots,X_k)=\sum_(-1)^jd(\alpha(X_0,\ldots,\hat_j,\ldots,X_k))(X_j) + \sum_(-1)^\alpha( _i,X_jX_0,\ldots,\hat_i,\ldots,\hat_j,\ldots,X_k)

:$(d\alpha)(X_1,\ldots,X_k) =\sum_^k(-1)^(\nabla_\alpha)(X_1,\ldots,\hat_i,\ldots,X_k)$

:$(\delta\alpha)(X_1,\ldots,X_)=-\sum_^n(\iota_(\nabla_\alpha))(X_1,\ldots,\hat_i,\ldots,X_k)$ given a positively oriented orthonormal frame $E_1,\ldots,E_n$.

:$(\mathcal_Y\alpha)(X_1,\ldots,X_k) =(\nabla_Y\alpha)(X_1,\ldots,X_k) - \sum_^k\alpha(X_1,\ldots,\nabla_Y,\ldots,X_k)$

Hodge decomposition

If $\partial M =\empty$, $\omega\in\Omega^k(M) \Rightarrow \exists \alpha\in\Omega^, \ \beta\in\Omega^, \ \gamma\in\Omega^k(M), \ d\gamma=0, \ \delta\gamma = 0$ such that

:$\omega = d\alpha + \delta\beta + \gamma$

Poincaré lemma

If a boundaryless manifold $M$ has trivial cohomology $H^k(M)=\$, then any closed $\omega\in\Omega^k(M)$ is exact. This is the case if ''M'' is contractible.

Relations to vector calculus

Identities in Euclidean 3-space

Let Euclidean metric $g(X,Y):=\langle X,Y\rangle = X\cdot Y$.

We use $\nabla = \left( , , \right)$ differential operator $\mathbb^3$

:$\iota_X\alpha = g(X,\alpha^) = X\cdot \alpha^$ for $\alpha\in\Omega^1(M)$.

:$\mathbf(X,Y,Z)=\langle X,Y\times Z\rangle = \langle X\times Y,Z\rangle$ ( ''scalar triple product'' )

:$X\times Y = ((X^\wedge Y^))^$ ( ''cross product'' )

:$\iota_X\alpha=-(X\times A)^$ if $\alpha\in\Omega^2(M),\ A=(\alpha)^$
:$X\cdot Y = (X^\wedge Y^)$ ( ''scalar product'' )

:$\nabla f=(df)^$ ( ''gradient'' )

:$X\cdot\nabla f=df(X)$ ( ''directional derivative'' )

:$\nabla\cdot X = d X^ = \delta X^$ ( ''divergence'' )

:$\nabla\times X = ( d X^)^$ ( ''curl'' )

:$\langle X,N\rangle\sigma = X^\flat$ where $N$ is the unit normal vector of $\partial M$ and $\sigma=\iota_\mathbf$ is the area form on $\partial M$.

:$\int_ d X^ = \int_ X^ = \int_\langle X,N\rangle\sigma$ ( '' divergence theorem'' )

Lie derivatives

:$\mathcal_X f =X\cdot \nabla f$ ( ''$0$-forms'' )

:$\mathcal_X \alpha = (\nabla_X\alpha^)^ +g(\alpha^,\nabla X)$ ( ''$1$-forms'' )

:$\mathcal_X\beta = \left( \nabla_XB - \nabla_BX + (\textX)B \right)^$ if $B=(\beta)^$ ( ''$2$-forms on $3$-manifolds'' )

:$\mathcal_X\rho = dq(X)+(\textX)q$ if $\rho= q \in \Omega^0(M)$ ( ''$n$-forms'' )

:$\mathcal_X(\mathbf)=(\text(X))\mathbf$

References

{{Reflist

Calculus
Mathematical identities
Mathematics-related lists
Differential forms
Differential operators
Generalizations of the derivative