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

$M$, $N$ are $n$-dimensional smooth manifolds, where $n\in \mathbb{\mathbb{N}}$. 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 $\mathrm{\Sigma }\subset M$.

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 $X,Y,Z\in \mathrm{\Gamma }(TM)$ 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 $\alpha ,\beta \in \mathrm{\Gamma }(T^{*}M)$ such that at a point $p\in M$ we have $\alpha {|}_p,\beta {|}_p\in T_p^{*}M$. An alternative notation for $\mathrm{\Gamma }(T^{*}M)$ is ${\mathrm{\Omega }}^1(M)$.

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 ${\mathrm{\Omega }}^k(M)$. For $0\le k,l,m\le n$ we usually write $\alpha \in {\mathrm{\Omega }}^k(M)$, $\beta \in {\mathrm{\Omega }}^l(M)$, $\gamma \in {\mathrm{\Omega }}^m(M)$.

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

Omitted elements of a sequence

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

$\alpha (X_0,\dots ,{\hat{X}}_i,\dots ,X_k):=\alpha (X_0,\dots ,X_{i-1},X_{i+1},\dots ,X_k).$

Exterior product

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

$(\alpha \wedge \beta )(X_1,\dots ,X_{k+l})=\sum _{\sigma \in S(k,k+l)}\text{sign}(\sigma )\alpha (X_{\sigma (1)},\dots ,X_{\sigma (k)})\otimes \beta (X_{\sigma (k+1)},\dots ,X_{\sigma (k+l)}).$

Directional derivative

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

Exterior derivative

The exterior derivative $d_k:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M)$ is defined for all $0\le k\le n$. We generally omit the subscript when it is clear from the context.

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

For $0<k\le n$,

$(d_k\omega )(X_0,\dots ,X_k)=\sum _{0\le j\le k}(-1{)}^j{d}_0(\omega (X_0,\dots ,{\hat{X}}_j,\dots ,X_k))(X_j)+\sum _{0\le i<j\le k}(-1{)}^{i+j}\omega ([X_i,X_j],X_0,\dots ,{\hat{X}}_i,\dots ,{\hat{X}}_j,\dots ,X_k).$

Lie bracket

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

$\mathrm{\forall }f\in {\mathrm{\Omega }}^0(M)\Rightarrow {\partial }_{[X,Y]}f={\partial }_X{\partial }_{Y}f-{\partial }_Y{\partial }_{X}f.$

Tangent maps

If $\varphi :M\to N$ is a smooth map, then $d\varphi {|}_p:T_{p}M\to T_{\varphi (p)}N$ defines a tangent map from $M$ to $N$. It is defined through curves $\gamma$ on $M$ with derivative ${\gamma }^{\prime }(0)=X\in T_{p}M$ such that

$d\varphi (X):=(\varphi \circ \gamma {)}^{\prime }.$

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

Pull-back

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

$\underset{\mathrm{\Sigma }}{\int }{\varphi }^{*}\alpha =\underset{\varphi (\mathrm{\Sigma })}{\int }\alpha .$

The pull-back can also be expressed as

$({\varphi }^{*}\alpha )(X_1,\dots ,X_k)=\alpha (d\varphi (X_1),\dots ,d\varphi (X_k)).$

Interior product

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

$({\iota }_Y\alpha )(X_1,\dots ,X_k)=\alpha (Y,X_1,\dots ,X_k).$

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 $s=\mathrm{sign}(g)$ 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 $\mathrm{♭}$ and sharp $\mathrm{♯}$. A section $A\in \mathrm{\Gamma }(TM)$ corresponds to the unique one-form $A^{\mathrm{♭}}\in {\mathrm{\Omega }}^1(M)$ such that for all sections $X\in \mathrm{\Gamma }(TM)$, we have:

$A^{\mathrm{♭}}(X)=g(A,X).$

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

$\alpha (X)=g({\alpha }^{\mathrm{♯}},X).$

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

$(A_1\wedge A_2\wedge \cdots \wedge A_k{)}^{\mathrm{♭}}=A_1^{\mathrm{♭}}\wedge A_2^{\mathrm{♭}}\wedge \cdots \wedge A_k^{\mathrm{♭}}$

$({\alpha }_1\wedge {\alpha }_2\wedge \cdots \wedge {\alpha }_k{)}^{\mathrm{♯}}={\alpha }_1^{\mathrm{♯}}\wedge {\alpha }_2^{\mathrm{♯}}\wedge \cdots \wedge {\alpha }_k^{\mathrm{♯}}.$

Hodge star

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

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

$(\star \alpha )(X_1,\dots ,X_{n-k})=\alpha (X_{n-k+1},\dots ,X_n).$

Co-differential operator

The co-differential operator $\delta :{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k-1}(M)$ on an $n$ dimensional manifold $M$ is defined by

$\delta :=(-1{)}^k{\star }^{-1}d\star =(-1{)}^{nk+n+1}\star d\star .$

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

Oriented manifold

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

Volume form

On an orientable manifold $M$ the canonical choice of a volume form given a metric tensor $g$ and an orientation is $\mathbf{𝐝}\mathbf{𝐞}\mathbf{𝐭}:=\sqrt{|\det g|}d{X}_1^{\mathrm{♭}}\wedge \dots \wedge d{X}_n^{\mathrm{♭}}$ for any basis $d{X}_1,\dots ,d{X}_n$ ordered to match the orientation.

Area form

Given a volume form $\mathbf{𝐝}\mathbf{𝐞}\mathbf{𝐭}$ and a unit normal vector $N$ we can also define an area form $\sigma :={\iota }_N\text{<mrow data-mjx-texclass="ORD">det</mrow>}$ on the boundary $\partial M.$

Bilinear form on k-forms

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

$⟨\alpha ,\beta ⟩{|}_p:=\star (\alpha \wedge \star \beta ){|}_p.$

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

$⟨⟨\alpha ,\beta ⟩⟩:=\underset{M}{\int }\alpha \wedge \star \beta .$

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

Lie derivative

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

${{ℒ}}_X=d\circ {\iota }_X+{\iota }_X\circ d.$

It describes the change of a $k$-form along a flow ${\varphi }_t$ associated to the section $X$.

Laplace–Beltrami operator

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

Important definitions

Definitions on Ω^k(M)

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

• closed if $d\alpha =0$
• exact if $\alpha =d\beta$ for some $\beta \in {\mathrm{\Omega }}^{k-1}$
• coclosed if $\delta \alpha =0$
• coexact if $\alpha =\delta \beta$ for some $\beta \in {\mathrm{\Omega }}^{k+1}$
• harmonic if closed and coclosed

Cohomology

The $k$-th cohomology of a manifold $M$ and its exterior derivative operators $d_0,\dots ,d_{n-1}$ is given by

$H^k(M):=\frac{\text{ker}(d_k)}{\text{im}(d_{k-1})}$

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

$[\alpha ]=[\beta ]⟺\alpha -\beta =d\eta \text{ for some }\eta \in {\mathrm{\Omega }}^{k-1}(M)$

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

Dirichlet energy

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

${{ℰ}}_{\text{D}}(\alpha ):=\frac{1}{2}⟨⟨d\alpha ,d\alpha ⟩⟩+\frac{1}{2}⟨⟨\delta \alpha ,\delta \alpha ⟩⟩$

Properties

Exterior derivative properties

$\underset{\mathrm{\Sigma }}{\int }d\alpha =\underset{\partial \mathrm{\Sigma }}{\int }\alpha$ ( Stokes' theorem )

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

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

$df(X)={\partial }_{X}f$ for $f\in {\mathrm{\Omega }}^0(M),X\in \mathrm{\Gamma }(TM)$ ( directional derivative )

$d\alpha =0$ for $\alpha \in {\mathrm{\Omega }}^n(M),\text{dim}(M)=n$

Exterior product properties

$\alpha \wedge \beta =(-1{)}^{kl}\beta \wedge \alpha$ for $\alpha \in {\mathrm{\Omega }}^k(M),\beta \in {\mathrm{\Omega }}^l(M)$ ( alternating )

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

$(\lambda \alpha )\wedge \beta =\lambda (\alpha \wedge \beta )$ for $\lambda \in \mathbb{ℝ}$ ( 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 $\alpha \in {\mathrm{\Omega }}^k(M)$ when $k$ is odd or $\mathrm{rank}\alpha \le 1$. 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

$d({\varphi }^{*}\alpha )={\varphi }^{*}(d\alpha )$ ( commutative with $d$ )

${\varphi }^{*}(\alpha \wedge \beta )=({\varphi }^{*}\alpha )\wedge ({\varphi }^{*}\beta )$ ( distributes over $\wedge$ )

$({\varphi }_1\circ {\varphi }_2{)}^{*}={\varphi }_2^{*}{\varphi }_1^{*}$ ( contravariant )

${\varphi }^{*}f=f\circ \varphi$ for $f\in {\mathrm{\Omega }}^0(N)$ ( function composition )

Musical isomorphism properties

$(X^{\mathrm{♭}}{)}^{\mathrm{♯}}=X$

$({\alpha }^{\mathrm{♯}}{)}^{\mathrm{♭}}=\alpha$

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 $\alpha \in {\mathrm{\Omega }}^k(M),\beta \in {\mathrm{\Omega }}^l(M)$ ( Leibniz rule )

${\iota }_X\alpha =\alpha (X)$ for $\alpha \in {\mathrm{\Omega }}^1(M)$

${\iota }_{X}f=0$ for $f\in {\mathrm{\Omega }}^0(M)$

${\iota }_X(f\alpha )=f{\iota }_X\alpha$ for $f\in {\mathrm{\Omega }}^0(M)$

Hodge star properties

$\star ({\lambda }_1\alpha +{\lambda }_2\beta )={\lambda }_1(\star \alpha )+{\lambda }_2(\star \beta )$ for ${\lambda }_1,{\lambda }_2\in \mathbb{ℝ}$ ( linearity )

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

${\star }^{(-1)}=s(-1{)}^{k(n-k)}\star$ ( inversion )

$\star (f\alpha )=f(\star \alpha )$ for $f\in {\mathrm{\Omega }}^0(M)$ ( commutative with $0$-forms )

$⟨⟨\alpha ,\alpha ⟩⟩=⟨⟨\star \alpha ,\star \alpha ⟩⟩$ for $\alpha \in {\mathrm{\Omega }}^1(M)$ ( Hodge star preserves $1$-form norm )

$\star \mathbf{𝟏}=\mathbf{𝐝}\mathbf{𝐞}\mathbf{𝐭}$ ( Hodge dual of constant function 1 is the volume form )

Co-differential operator properties

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

$\star \delta =(-1{)}^{k}d\star$ and $\star d=(-1{)}^{k+1}\delta \star$ ( Hodge adjoint to $d$ )

$⟨⟨d\alpha ,\beta ⟩⟩=⟨⟨\alpha ,\delta \beta ⟩⟩$ if $\partial M=0$ ( $\delta$ adjoint to $d$ )

In general, $\underset{M}{\int }d\alpha \wedge \star \beta =\underset{\partial M}{\int }\alpha \wedge \star \beta +\underset{M}{\int }\alpha \wedge \star \delta \beta$

$\delta f=0$ for $f\in {\mathrm{\Omega }}^0(M)$

Lie derivative properties

$d\circ {{ℒ}}_X={{ℒ}}_X\circ d$ ( commutative with $d$ )

${\iota }_X\circ {{ℒ}}_X={{ℒ}}_X\circ {\iota }_X$ ( commutative with ${\iota }_X$ )

${{ℒ}}_X({\iota }_Y\alpha )={\iota }_{[X,Y]}\alpha +{\iota }_Y{{ℒ}}_X\alpha$

${{ℒ}}_X(\alpha \wedge \beta )=({{ℒ}}_X\alpha )\wedge \beta +\alpha \wedge ({{ℒ}}_X\beta )$ ( Leibniz rule )

Exterior calculus identities

${\iota }_X(\star \mathbf{𝟏})=\star X^{\mathrm{♭}}$

${\iota }_X(\star \alpha )=(-1{)}^k\star (X^{\mathrm{♭}}\wedge \alpha )$ if $\alpha \in {\mathrm{\Omega }}^k(M)$

${\iota }_X({\varphi }^{*}\alpha )={\varphi }^{*}({\iota }_{d\varphi (X)}\alpha )$

$\nu ,\mu \in {\mathrm{\Omega }}^n(M),\mu \text{ non-zero }\Rightarrow \mathrm{\exists }f\in {\mathrm{\Omega }}^0(M):\nu =f\mu$

$X^{\mathrm{♭}}\wedge \star Y^{\mathrm{♭}}=g(X,Y)(\star \mathbf{𝟏})$ ( bilinear form )

$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$ ( Jacobi identity )

Dimensions

If $n=\dim M$

$\dim {\mathrm{\Omega }}^k(M)=(\genfrac{}{}{0ex}{}{n}{k})$ for $0\le k\le n$

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

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

$\{X_{\sigma (1)}^{\mathrm{♭}}\wedge \dots \wedge X_{\sigma (k)}^{\mathrm{♭}}:\sigma \in S(k,n)\}$

Exterior products

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

$\alpha (X)=\det \left[\begin{array}{c}\alpha (X)\\ \end{array}\right]$

$(\alpha \wedge \beta )(X,Y)=\det \left[\begin{array}{cc}\alpha (X)& \alpha (Y)\\ \beta (X)& \beta (Y)\\ \end{array}\right]$

$(\alpha \wedge \beta \wedge \gamma )(X,Y,Z)=\det \left[\begin{array}{ccc}\alpha (X)& \alpha (Y)& \alpha (Z)\\ \beta (X)& \beta (Y)& \beta (Z)\\ \gamma (X)& \gamma (Y)& \gamma (Z)\end{array}\right]$

$({\alpha }_1\wedge \dots \wedge {\alpha }_l)(X_1,\dots ,X_l)=\det \left[\begin{array}{cccc}{\alpha }_1(X_1)& {\alpha }_1(X_2)& \dots & {\alpha }_1(X_l)\\ {\alpha }_2(X_1)& {\alpha }_2(X_2)& \dots & {\alpha }_2(X_l)\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_l(X_1)& {\alpha }_l(X_2)& \dots & {\alpha }_l(X_l)\end{array}\right]$

Projection and rejection

$(-1{)}^k{\iota }_X\star \alpha =\star (X^{\mathrm{♭}}\wedge \alpha )$ ( interior product ${\iota }_X\star$ dual to wedge $X^{\mathrm{♭}}\wedge$ )

$({\iota }_X\alpha )\wedge \star \beta =\alpha \wedge \star (X^{\mathrm{♭}}\wedge \beta )$ for $\alpha \in {\mathrm{\Omega }}^{k+1}(M),\beta \in {\mathrm{\Omega }}^k(M)$

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

• ${\iota }_X\circ (X^{\mathrm{♭}}\wedge ):{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^k(M)$ is the projection of $\alpha$ onto the orthogonal complement of $X$.
• $(X^{\mathrm{♭}}\wedge )\circ {\iota }_X:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^k(M)$ is the rejection of $\alpha$, the remainder of the projection.
• thus ${\iota }_X\circ (X^{\mathrm{♭}}\wedge )+(X^{\mathrm{♭}}\wedge )\circ {\iota }_X=\text{id}$ ( projection–rejection decomposition )

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

• $\mathbf{𝐭}:={\iota }_N\circ (N^{\mathrm{♭}}\wedge )$ extracts the tangential component of the boundary.
• $\mathbf{𝐧}:=(\text{id}-\mathbf{𝐭})$ extracts the normal component of the boundary.

Sum expressions

$(d\alpha )(X_0,\dots ,X_k)=\sum _{0\le j\le k}(-1{)}^{j}d(\alpha (X_0,\dots ,{\hat{X}}_j,\dots ,X_k))(X_j)+\sum _{0\le i<j\le k}(-1{)}^{i+j}\alpha ([X_i,X_j],X_0,\dots ,{\hat{X}}_i,\dots ,{\hat{X}}_j,\dots ,X_k)$

$(d\alpha )(X_1,\dots ,X_k)=\sum _{i=1}^k(-1{)}^{i+1}({\mathrm{\nabla }}_{X_i}\alpha )(X_1,\dots ,{\hat{X}}_i,\dots ,X_k)$

$(\delta \alpha )(X_1,\dots ,X_{k-1})=-\sum _{i=1}^n({\iota }_{E_i}({\mathrm{\nabla }}_{E_i}\alpha ))(X_1,\dots ,{\hat{X}}_i,\dots ,X_k)$ given a positively oriented orthonormal frame $E_1,\dots ,E_n$.

$({{ℒ}}_Y\alpha )(X_1,\dots ,X_k)=({\mathrm{\nabla }}_Y\alpha )(X_1,\dots ,X_k)-\sum _{i=1}^k\alpha (X_1,\dots ,{\mathrm{\nabla }}_{X_i}Y,\dots ,X_k)$

Hodge decomposition

If $\partial M=\mathrm{\varnothing }$, $\omega \in {\mathrm{\Omega }}^k(M)\Rightarrow \mathrm{\exists }\alpha \in {\mathrm{\Omega }}^{k-1},\beta \in {\mathrm{\Omega }}^{k+1},\gamma \in {\mathrm{\Omega }}^k(M),d\gamma =0,\delta \gamma =0$ such that

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

Poincaré lemma

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

Relations to vector calculus

Identities in Euclidean 3-space

Let Euclidean metric $g(X,Y):=⟨X,Y⟩=X\cdot Y$.

We use $\mathrm{\nabla }=(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z})$ differential operator ${\mathbb{ℝ}}^3$

${\iota }_X\alpha =g(X,{\alpha }^{\mathrm{♯}})=X\cdot {\alpha }^{\mathrm{♯}}$ for $\alpha \in {\mathrm{\Omega }}^1(M)$.

$\mathbf{𝐝}\mathbf{𝐞}\mathbf{𝐭}(X,Y,Z)=⟨X,Y\times Z⟩=⟨X\times Y,Z⟩$ ( scalar triple product )

$X\times Y=(\star (X^{\mathrm{♭}}\wedge Y^{\mathrm{♭}}){)}^{\mathrm{♯}}$ ( cross product )

${\iota }_X\alpha =-(X\times A{)}^{\mathrm{♭}}$ if $\alpha \in {\mathrm{\Omega }}^2(M),A=(\star \alpha {)}^{\mathrm{♯}}$

$X\cdot Y=\star (X^{\mathrm{♭}}\wedge \star Y^{\mathrm{♭}})$ ( scalar product )

$\mathrm{\nabla }f=(df{)}^{\mathrm{♯}}$ ( gradient )

$X\cdot \mathrm{\nabla }f=df(X)$ ( directional derivative )

$\mathrm{\nabla }\cdot X=\star d\star X^{\mathrm{♭}}=-\delta X^{\mathrm{♭}}$ ( divergence )

$\mathrm{\nabla }\times X=(\star d{X}^{\mathrm{♭}}{)}^{\mathrm{♯}}$ ( curl )

$⟨X,N⟩\sigma =\star X^{\mathrm{♭}}$ where $N$ is the unit normal vector of $\partial M$ and $\sigma ={\iota }_N\mathbf{𝐝}\mathbf{𝐞}\mathbf{𝐭}$ is the area form on $\partial M$.

$\underset{\mathrm{\Sigma }}{\int }d\star X^{\mathrm{♭}}=\underset{\partial \mathrm{\Sigma }}{\int }\star X^{\mathrm{♭}}=\underset{\partial \mathrm{\Sigma }}{\int }⟨X,N⟩\sigma$ ( divergence theorem )

Lie derivatives

${{ℒ}}_{X}f=X\cdot \mathrm{\nabla }f$ ( $0$-forms )

${{ℒ}}_X\alpha =({\mathrm{\nabla }}_X{\alpha }^{\mathrm{♯}}{)}^{\mathrm{♭}}+g({\alpha }^{\mathrm{♯}},\mathrm{\nabla }X)$ ( $1$-forms )

$\star {{ℒ}}_X\beta ={({\mathrm{\nabla }}_{X}B-{\mathrm{\nabla }}_{B}X+(\text{div}X)B)}^{\mathrm{♭}}$ if $B=(\star \beta {)}^{\mathrm{♯}}$ ( $2$-forms on $3$-manifolds )

$\star {{ℒ}}_X\rho =dq(X)+(\text{div}X)q$ if $\rho =\star q\in {\mathrm{\Omega }}^0(M)$ ( $n$-forms )

${{ℒ}}_X(\mathbf{𝐝}\mathbf{𝐞}\mathbf{𝐭})=(\text{div}(X))\mathbf{𝐝}\mathbf{𝐞}\mathbf{𝐭}$