Exterior calculus identities

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

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

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

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

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

Tangent and cotangent bundles

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

${\displaystyle T_{p}M}$, ${\displaystyle T_{q}N}$ denote the tangent spaces of ${\displaystyle M}$, ${\displaystyle N}$ at the points ${\displaystyle p}$, ${\displaystyle q}$, respectively. ${\displaystyle T_{p}^{*}M}$ denotes the cotangent space of ${\displaystyle M}$ at the point ${\displaystyle p}$.

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

Differential k-forms

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

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

Omitted elements of a sequence

When we are given ${\displaystyle (k+1)}$ inputs ${\displaystyle X_{0},\ldots ,X_{k}}$ and a ${\displaystyle k}$-form ${\displaystyle \alpha \in \Omega ^{k}(M)}$ we denote omission of the ${\displaystyle i}$th entry by writing

${\displaystyle \alpha (X_{0},\ldots ,{\hat {X}}_{i},\ldots ,X_{k}):=\alpha (X_{0},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{k}).}$

Exterior product

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

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

Directional derivative

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

Exterior derivative

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

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

For ${\displaystyle 0<k\leq n}$,^[6]

${\displaystyle (d_{k}\omega )(X_{0},\ldots ,X_{k})=\sum _{0\leq j\leq k}(-1)^{j}d_{0}(\omega (X_{0},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}))(X_{j})+\sum _{0\leq i<j\leq k}(-1)^{i+j}\omega ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}).}$

Lie bracket

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

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

Tangent maps

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

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

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

Pull-back

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

${\displaystyle \int _{\Sigma }\phi ^{*}\alpha =\int _{\phi (\Sigma )}\alpha .}$

The pull-back can also be expressed as

${\displaystyle (\phi ^{*}\alpha )(X_{1},\ldots ,X_{k})=\alpha (d\phi (X_{1}),\ldots ,d\phi (X_{k})).}$

Interior product

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

${\displaystyle (\iota _{Y}\alpha )(X_{1},\ldots ,X_{k})=\alpha (Y,X_{1},\ldots ,X_{k}).}$

Metric tensor

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

Musical isomorphisms

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

${\displaystyle A^{\flat }(X)=g(A,X).}$

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

${\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).}$

These mappings extend via multilinearity to mappings from ${\displaystyle k}$-vector fields to ${\displaystyle k}$-forms and ${\displaystyle k}$-forms to ${\displaystyle k}$-vector fields through

${\displaystyle (A_{1}\wedge A_{2}\wedge \cdots \wedge A_{k})^{\flat }=A_{1}^{\flat }\wedge A_{2}^{\flat }\wedge \cdots \wedge A_{k}^{\flat }}$

${\displaystyle (\alpha _{1}\wedge \alpha _{2}\wedge \cdots \wedge \alpha _{k})^{\sharp }=\alpha _{1}^{\sharp }\wedge \alpha _{2}^{\sharp }\wedge \cdots \wedge \alpha _{k}^{\sharp }.}$

Hodge star

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

It can be defined in terms of an oriented frame ${\displaystyle (X_{1},\ldots ,X_{n})}$ for ${\displaystyle TM}$, orthonormal with respect to the given metric tensor ${\displaystyle g}$:

${\displaystyle ({\star }\alpha )(X_{1},\ldots ,X_{n-k})=\alpha (X_{n-k+1},\ldots ,X_{n}).}$

Co-differential operator

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

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

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

Oriented manifold

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

Volume form

On an orientable manifold ${\displaystyle M}$ the canonical choice of a volume form given a metric tensor ${\displaystyle g}$ and an orientation is ${\displaystyle \mathbf {det} :={\sqrt {|\det g|}}\;dX_{1}^{\flat }\wedge \ldots \wedge dX_{n}^{\flat }}$ for any basis ${\displaystyle dX_{1},\ldots ,dX_{n}}$ ordered to match the orientation.

Area form

Given a volume form ${\displaystyle \mathbf {det} }$ and a unit normal vector ${\displaystyle N}$ we can also define an area form ${\displaystyle \sigma :=\iota _{N}{\textbf {det}}}$ on the boundary ${\displaystyle \partial M.}$

Bilinear form on k-forms

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

${\displaystyle \langle \alpha ,\beta \rangle |_{p}:={\star }(\alpha \wedge {\star }\beta )|_{p}.}$

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

${\displaystyle \langle \!\langle \alpha ,\beta \rangle \!\rangle :=\int _{M}\alpha \wedge {\star }\beta .}$

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

Lie derivative

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

${\displaystyle {\mathcal {L}}_{X}=d\circ \iota _{X}+\iota _{X}\circ d.}$

It describes the change of a ${\displaystyle k}$-form along a flow ${\displaystyle \phi _{t}}$ associated to the section ${\displaystyle X}$.

Laplace–Beltrami operator

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

Important definitions

Definitions on Ω^k(M)

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

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

Cohomology

The ${\displaystyle k}$-th cohomology of a manifold ${\displaystyle M}$ and its exterior derivative operators ${\displaystyle d_{0},\ldots ,d_{n-1}}$ is given by

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

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

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

A closed surface of genus ${\displaystyle g}$ will have ${\displaystyle 2g}$ generators which are harmonic.

Dirichlet energy

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

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

Properties

Exterior derivative properties

${\displaystyle \int _{\Sigma }d\alpha =\int _{\partial \Sigma }\alpha }$ ( Stokes' theorem )

${\displaystyle d\circ d=0}$ ( cochain complex )

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

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

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

Exterior product properties

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

${\displaystyle (\alpha \wedge \beta )\wedge \gamma =\alpha \wedge (\beta \wedge \gamma )}$ ( associativity )

${\displaystyle (\lambda \alpha )\wedge \beta =\lambda (\alpha \wedge \beta )}$ for ${\displaystyle \lambda \in \mathbb {R} }$ ( compatibility of scalar multiplication )

${\displaystyle \alpha \wedge (\beta _{1}+\beta _{2})=\alpha \wedge \beta _{1}+\alpha \wedge \beta _{2}}$ ( distributivity over addition )

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

Pull-back properties

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

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

${\displaystyle (\phi _{1}\circ \phi _{2})^{*}=\phi _{2}^{*}\phi _{1}^{*}}$ ( contravariant )

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

Musical isomorphism properties

${\displaystyle (X^{\flat })^{\sharp }=X}$

${\displaystyle (\alpha ^{\sharp })^{\flat }=\alpha }$

Interior product properties

${\displaystyle \iota _{X}\circ \iota _{X}=0}$ ( nilpotent )

${\displaystyle \iota _{X}\circ \iota _{Y}=-\iota _{Y}\circ \iota _{X}}$

${\displaystyle \iota _{X}(\alpha \wedge \beta )=(\iota _{X}\alpha )\wedge \beta +(-1)^{k}\alpha \wedge (\iota _{X}\beta )}$ for ${\displaystyle \alpha \in \Omega ^{k}(M),\ \beta \in \Omega ^{l}(M)}$ ( Leibniz rule )

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

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

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

Hodge star properties

${\displaystyle {\star }(\lambda _{1}\alpha +\lambda _{2}\beta )=\lambda _{1}({\star }\alpha )+\lambda _{2}({\star }\beta )}$ for ${\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {R} }$ ( linearity )

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

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

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

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

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

Co-differential operator properties

${\displaystyle \delta \circ \delta =0}$ ( nilpotent )

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

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

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

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

Lie derivative properties

${\displaystyle d\circ {\mathcal {L}}_{X}={\mathcal {L}}_{X}\circ d}$ ( commutative with ${\displaystyle d}$ )

${\displaystyle \iota _{X}\circ {\mathcal {L}}_{X}={\mathcal {L}}_{X}\circ \iota _{X}}$ ( commutative with ${\displaystyle \iota _{X}}$ )

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

${\displaystyle {\mathcal {L}}_{X}(\alpha \wedge \beta )=({\mathcal {L}}_{X}\alpha )\wedge \beta +\alpha \wedge ({\mathcal {L}}_{X}\beta )}$ ( Leibniz rule )

Exterior calculus identities

${\displaystyle \iota _{X}({\star }\mathbf {1} )={\star }X^{\flat }}$

${\displaystyle \iota _{X}({\star }\alpha )=(-1)^{k}{\star }(X^{\flat }\wedge \alpha )}$ if ${\displaystyle \alpha \in \Omega ^{k}(M)}$

${\displaystyle \iota _{X}(\phi ^{*}\alpha )=\phi ^{*}(\iota _{d\phi (X)}\alpha )}$

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

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

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

Dimensions

If ${\displaystyle n=\dim M}$

${\displaystyle \dim \Omega ^{k}(M)={\binom {n}{k}}}$ for ${\displaystyle 0\leq k\leq n}$

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

If ${\displaystyle X_{1},\ldots ,X_{n}\in \Gamma (TM)}$ is a basis, then a basis of ${\displaystyle \Omega ^{k}(M)}$ is

${\displaystyle \{X_{\sigma (1)}^{\flat }\wedge \ldots \wedge X_{\sigma (k)}^{\flat }\ :\ \sigma \in S(k,n)\}}$

Exterior products

Let ${\displaystyle \alpha ,\beta ,\gamma ,\alpha _{i}\in \Omega ^{1}(M)}$ and ${\displaystyle X,Y,Z,X_{i}}$ be vector fields.

${\displaystyle \alpha (X)=\det {\begin{bmatrix}\alpha (X)\\\end{bmatrix}}}$

${\displaystyle (\alpha \wedge \beta )(X,Y)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)\\\beta (X)&\beta (Y)\\\end{bmatrix}}}$

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

${\displaystyle (\alpha _{1}\wedge \ldots \wedge \alpha _{l})(X_{1},\ldots ,X_{l})=\det {\begin{bmatrix}\alpha _{1}(X_{1})&\alpha _{1}(X_{2})&\dots &\alpha _{1}(X_{l})\\\alpha _{2}(X_{1})&\alpha _{2}(X_{2})&\dots &\alpha _{2}(X_{l})\\\vdots &\vdots &\ddots &\vdots \\\alpha _{l}(X_{1})&\alpha _{l}(X_{2})&\dots &\alpha _{l}(X_{l})\end{bmatrix}}}$

Projection and rejection

${\displaystyle (-1)^{k}\iota _{X}{\star }\alpha ={\star }(X^{\flat }\wedge \alpha )}$ ( interior product ${\displaystyle \iota _{X}{\star }}$ dual to wedge ${\displaystyle X^{\flat }\wedge }$ )

${\displaystyle (\iota _{X}\alpha )\wedge {\star }\beta =\alpha \wedge {\star }(X^{\flat }\wedge \beta )}$ for ${\displaystyle \alpha \in \Omega ^{k+1}(M),\beta \in \Omega ^{k}(M)}$

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

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

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

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

Sum expressions

${\displaystyle (d\alpha )(X_{0},\ldots ,X_{k})=\sum _{0\leq j\leq k}(-1)^{j}d(\alpha (X_{0},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}))(X_{j})+\sum _{0\leq i<j\leq k}(-1)^{i+j}\alpha ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k})}$

${\displaystyle (d\alpha )(X_{1},\ldots ,X_{k})=\sum _{i=1}^{k}(-1)^{i+1}(\nabla _{X_{i}}\alpha )(X_{1},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})}$

${\displaystyle (\delta \alpha )(X_{1},\ldots ,X_{k-1})=-\sum _{i=1}^{n}(\iota _{E_{i}}(\nabla _{E_{i}}\alpha ))(X_{1},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})}$ given a positively oriented orthonormal frame ${\displaystyle E_{1},\ldots ,E_{n}}$.

${\displaystyle ({\mathcal {L}}_{Y}\alpha )(X_{1},\ldots ,X_{k})=(\nabla _{Y}\alpha )(X_{1},\ldots ,X_{k})-\sum _{i=1}^{k}\alpha (X_{1},\ldots ,\nabla _{X_{i}}Y,\ldots ,X_{k})}$

Hodge decomposition

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

${\displaystyle \omega =d\alpha +\delta \beta +\gamma }$

Poincaré lemma

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

Relations to vector calculus

Identities in Euclidean 3-space

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

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

${\displaystyle \iota _{X}\alpha =g(X,\alpha ^{\sharp })=X\cdot \alpha ^{\sharp }}$ for ${\displaystyle \alpha \in \Omega ^{1}(M)}$.

${\displaystyle \mathbf {det} (X,Y,Z)=\langle X,Y\times Z\rangle =\langle X\times Y,Z\rangle }$ ( scalar triple product )

${\displaystyle X\times Y=({\star }(X^{\flat }\wedge Y^{\flat }))^{\sharp }}$ ( cross product )

${\displaystyle \iota _{X}\alpha =-(X\times A)^{\flat }}$ if ${\displaystyle \alpha \in \Omega ^{2}(M),\ A=({\star }\alpha )^{\sharp }}$

${\displaystyle X\cdot Y={\star }(X^{\flat }\wedge {\star }Y^{\flat })}$ ( scalar product )

${\displaystyle \nabla f=(df)^{\sharp }}$ ( gradient )

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

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

${\displaystyle \nabla \times X=({\star }dX^{\flat })^{\sharp }}$ ( curl )

${\displaystyle \langle X,N\rangle \sigma ={\star }X^{\flat }}$ where ${\displaystyle N}$ is the unit normal vector of ${\displaystyle \partial M}$ and ${\displaystyle \sigma =\iota _{N}\mathbf {det} }$ is the area form on ${\displaystyle \partial M}$.

${\displaystyle \int _{\Sigma }d{\star }X^{\flat }=\int _{\partial \Sigma }{\star }X^{\flat }=\int _{\partial \Sigma }\langle X,N\rangle \sigma }$ ( divergence theorem )

Lie derivatives

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

${\displaystyle {\mathcal {L}}_{X}\alpha =(\nabla _{X}\alpha ^{\sharp })^{\flat }+g(\alpha ^{\sharp },\nabla X)}$ ( ${\displaystyle 1}$-forms )

${\displaystyle {\star }{\mathcal {L}}_{X}\beta =\left(\nabla _{X}B-\nabla _{B}X+({\text{div}}X)B\right)^{\flat }}$ if ${\displaystyle B=({\star }\beta )^{\sharp }}$ ( ${\displaystyle 2}$-forms on ${\displaystyle 3}$-manifolds )

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

${\displaystyle {\mathcal {L}}_{X}(\mathbf {det} )=({\text{div}}(X))\mathbf {det} }$

References