# Exterior calculus identities

This article summarizes several identities in exterior calculus, a mathematical notation used in differential geometry.

## Notation

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

### Manifold

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

### 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 $C^{\infty }(M)$ on $M$. $\mathbf {1} \in \Omega ^{0}(M)$ 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
    $\alpha (X_{0},\ldots ,{\hat {X}}_{i},\ldots ,X_{k}):=\alpha (X_{0},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{k}).$

### Exterior product

The exterior product is also known as the wedge product. It is denoted by $\wedge :\Omega ^{k}(M)\times \Omega ^{l}(M)\rightarrow \Omega ^{k+l}(M)$. 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 $\alpha \wedge \beta \in \Omega ^{k+l}(M)$. It can be written using the set $S(k,k+l)$ of all permutations $\sigma$ of $\{1,\ldots ,n\}$ such that $\sigma (1)<\ldots <\sigma (k),\ \sigma (k+1)<\ldots <\sigma (k+l)$ as
    $(\alpha \wedge \beta )(X_{1},\ldots ,X_{k+l})=\sum _{\sigma \in S(k,k+l)}{\text{sign}}(\sigma )\alpha (X_{\sigma (1)},\ldots ,X_{\sigma (k)})\otimes \beta (X_{\sigma (k+1)},\ldots ,X_{\sigma (k+l)}).$

### Directional derivative

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

### Exterior derivative

The exterior derivative $d_{k}:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M)$ 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 $d_{0}f\in \Omega ^{1}(M)$ as the $1$-form that gives the directional derivative, i.e., for the section $X\in \Gamma (TM)$ we have $(d_{0}f)(X)=\partial _{X}f$, the directional derivative of $f$ along $X$.
For $0<k\leq n$,
    $(d_{k}\omega )(X_{0},\ldots ,X_{k})=\sum _{0\leq j\leq k}(-1)^{j}d_{0}(\omega (X_{0},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}))(X_{j})+\sum _{0\leq i<j\leq k}(-1)^{i+j}\omega ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}).$

### Lie bracket

The Lie bracket of sections $X,Y\in \Gamma (TM)$ is defined as the unique section $[X,Y]\in \Gamma (TM)$ that satisfies
    $\forall f\in \Omega ^{0}(M)\Rightarrow \partial _{[X,Y]}f=\partial _{X}\partial _{Y}f-\partial _{Y}\partial _{X}f.$

### Tangent maps

If $\phi :M\rightarrow N$ is a smooth map, then $d\phi |_{p}:T_{p}M\rightarrow T_{\phi (p)}N$ defines a tangent map from $M$ to $N$. It is defined through curves $\gamma$ on $M$ with derivative $\gamma '(0)=X\in T_{p}M$ 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 $\Sigma \subset M$
    $\int _{\Sigma }\phi ^{*}\alpha =\int _{\phi (\Sigma )}\alpha .$
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 $\iota _{Y}:\Omega ^{k+1}(M)\rightarrow \Omega ^{k}(M)$ that effectively substitutes the first input of a $(k+1)$-form with $Y$. If $\alpha \in \Omega ^{k+1}(M)$ 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 $s=\operatorname {sign} (g)$ 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 $\flat$ and sharp $\sharp$. A section $A\in \Gamma (TM)$ corresponds to the unique one-form $A^{\flat }\in \Omega ^{1}(M)$ such that for all sections $X\in \Gamma (TM)$, we have:
    $A^{\flat }(X)=g(A,X).$
A one-form $\alpha \in \Omega ^{1}(M)$ corresponds to the unique vector field $\alpha ^{\sharp }\in \Gamma (TM)$ 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
    $(A_{1}\wedge A_{2}\wedge \cdots \wedge A_{k})^{\flat }=A_{1}^{\flat }\wedge A_{2}^{\flat }\wedge \cdots \wedge A_{k}^{\flat }$
    $(\alpha _{1}\wedge \alpha _{2}\wedge \cdots \wedge \alpha _{k})^{\sharp }=\alpha _{1}^{\sharp }\wedge \alpha _{2}^{\sharp }\wedge \cdots \wedge \alpha _{k}^{\sharp }.$

### Hodge star

For an n-manifold M, the Hodge star operator ${\star }:\Omega ^{k}(M)\rightarrow \Omega ^{n-k}(M)$ is a duality mapping taking a $k$-form $\alpha \in \Omega ^{k}(M)$ to an $(n{-}k)$-form $({\star }\alpha )\in \Omega ^{n-k}(M)$.
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 $g$:
    $({\star }\alpha )(X_{1},\ldots ,X_{n-k})=\alpha (X_{n-k+1},\ldots ,X_{n}).$

### Co-differential operator

The co-differential operator $\delta :\Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)$ 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 $\mu \in \Omega ^{n}(M)$ 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 $\mathbf {det} :={\sqrt {|\det g|}}\;dX_{1}^{\flat }\wedge \ldots \wedge dX_{n}^{\flat }$ for any basis $dX_{1},\ldots ,dX_{n}$ ordered to match the orientation.

### Area form

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

### Bilinear form on k-forms

A generalization of the metric tensor, the symmetric bilinear form between two $k$-forms $\alpha ,\beta \in \Omega ^{k}(M)$, is defined pointwise on $M$ by
    $\langle \alpha ,\beta \rangle |_{p}:={\star }(\alpha \wedge {\star }\beta )|_{p}.$
The $L^{2}$-bilinear form for the space of $k$-forms $\Omega ^{k}(M)$ is defined by
    $\langle \!\langle \alpha ,\beta \rangle \!\rangle :=\int _{M}\alpha \wedge {\star }\beta .$
In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).

### Lie derivative

We define the Lie derivative ${\mathcal {L}}:\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)$ through Cartan's magic formula for a given section $X\in \Gamma (TM)$ as
    ${\mathcal {L}}_{X}=d\circ \iota _{X}+\iota _{X}\circ d.$
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 $\beta \in \Omega ^{k-1}$
* coclosed if $\delta \alpha =0$
* coexact if $\alpha =\delta \beta$ for some $\beta \in \Omega ^{k+1}$
* harmonic if closed and coclosed

### Cohomology

The $k$-th cohomology of a manifold $M$ and its exterior derivative operators $d_{0},\ldots ,d_{n-1}$ is given by
    $H^{k}(M):={\frac {{\text{ker}}(d_{k})}{{\text{im}}(d_{k-1})}}$
Two closed $k$-forms $\alpha ,\beta \in \Omega ^{k}(M)$ are in the same cohomology class if their difference is an exact form i.e.
    $[\alpha ]=[\beta ]\ \ \Longleftrightarrow \ \ \alpha {-}\beta =d\eta \ {\text{ for some }}\eta \in \Omega ^{k-1}(M)$
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
    ${\mathcal {E}}_{\text{D}}(\alpha ):={\dfrac {1}{2}}\langle \!\langle d\alpha ,d\alpha \rangle \!\rangle +{\dfrac {1}{2}}\langle \!\langle \delta \alpha ,\delta \alpha \rangle \!\rangle$

## Properties
### Exterior derivative properties

    $\int _{\Sigma }d\alpha =\int _{\partial \Sigma }\alpha$ ( 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 $\alpha \in \Omega ^{n}(M),\ {\text{dim}}(M)=n$

### Exterior product properties

    $\alpha \wedge \beta =(-1)^{kl}\beta \wedge \alpha$ 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 $\lambda \in \mathbb {R}$ ( 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 $\alpha \in \Omega ^{k}(M)$ when $k$ is odd or $\operatorname {rank} \alpha \leq 1$. 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

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

### 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

    ${\star }(\lambda _{1}\alpha +\lambda _{2}\beta )=\lambda _{1}({\star }\alpha )+\lambda _{2}({\star }\beta )$ for $\lambda _{1},\lambda _{2}\in \mathbb {R}$ ( linearity )
    ${\star }{\star }\alpha =s(-1)^{k(n-k)}\alpha$ for $\alpha \in \Omega ^{k}(M)$, $n=\dim(M)$, and $s=\operatorname {sign} (g)$ the sign of the metric
    ${\star }^{(-1)}=s(-1)^{k(n-k)}{\star }$ ( inversion )
    ${\star }(f\alpha )=f({\star }\alpha )$ for $f\in \Omega ^{0}(M)$ ( commutative with $0$-forms )
    $\langle \!\langle \alpha ,\alpha \rangle \!\rangle =\langle \!\langle {\star }\alpha ,{\star }\alpha \rangle \!\rangle$ for $\alpha \in \Omega ^{1}(M)$ ( Hodge star preserves $1$-form norm )
    ${\star }\mathbf {1} =\mathbf {det}$ ( 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 $d$ )
    $\langle \!\langle d\alpha ,\beta \rangle \!\rangle =\langle \!\langle \alpha ,\delta \beta \rangle \!\rangle$ if $\partial M=0$ ( $\delta$ adjoint to $d$ )
    In general, $\int _{M}d\alpha \wedge \star \beta =\int _{\partial M}\alpha \wedge \star \beta +\int _{M}\alpha \wedge \star \delta \beta$
    $\delta f=0$ for $f\in \Omega ^{0}(M)$

### Lie derivative properties

    $d\circ {\mathcal {L}}_{X}={\mathcal {L}}_{X}\circ d$ ( commutative with $d$ )
    $\iota _{X}\circ {\mathcal {L}}_{X}={\mathcal {L}}_{X}\circ \iota _{X}$ ( commutative with $\iota _{X}$ )
    ${\mathcal {L}}_{X}(\iota _{Y}\alpha )=\iota _{[X,Y]}\alpha +\iota _{Y}{\mathcal {L}}_{X}\alpha$
    ${\mathcal {L}}_{X}(\alpha \wedge \beta )=({\mathcal {L}}_{X}\alpha )\wedge \beta +\alpha \wedge ({\mathcal {L}}_{X}\beta )$ ( Leibniz rule )

## Exterior calculus identities

    $\iota _{X}({\star }\mathbf {1} )={\star }X^{\flat }$
    $\iota _{X}({\star }\alpha )=(-1)^{k}{\star }(X^{\flat }\wedge \alpha )$ if $\alpha \in \Omega ^{k}(M)$
    $\iota _{X}(\phi ^{*}\alpha )=\phi ^{*}(\iota _{d\phi (X)}\alpha )$
    $\nu ,\mu \in \Omega ^{n}(M),\mu {\text{ non-zero }}\ \Rightarrow \ \exists \ f\in \Omega ^{0}(M):\ \nu =f\mu$
    $X^{\flat }\wedge {\star }Y^{\flat }=g(X,Y)({\star }\mathbf {1} )$ ( bilinear form )
    $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$ ( Jacobi identity )

### Dimensions

If $n=\dim M$
    $\dim \Omega ^{k}(M)={\binom {n}{k}}$ 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
    $\{X_{\sigma (1)}^{\flat }\wedge \ldots \wedge X_{\sigma (k)}^{\flat }\ :\ \sigma \in S(k,n)\}$

### Exterior products

Let $\alpha ,\beta ,\gamma ,\alpha _{i}\in \Omega ^{1}(M)$ and $X,Y,Z,X_{i}$ be vector fields.
    $\alpha (X)=\det {\begin{bmatrix}\alpha (X)\\\end{bmatrix}}$
    $(\alpha \wedge \beta )(X,Y)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)\\\beta (X)&\beta (Y)\\\end{bmatrix}}$
    $(\alpha \wedge \beta \wedge \gamma )(X,Y,Z)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)&\alpha (Z)\\\beta (X)&\beta (Y)&\beta (Z)\\\gamma (X)&\gamma (Y)&\gamma (Z)\end{bmatrix}}$
    $(\alpha _{1}\wedge \ldots \wedge \alpha _{l})(X_{1},\ldots ,X_{l})=\det {\begin{bmatrix}\alpha _{1}(X_{1})&\alpha _{1}(X_{2})&\dots &\alpha _{1}(X_{l})\\\alpha _{2}(X_{1})&\alpha _{2}(X_{2})&\dots &\alpha _{2}(X_{l})\\\vdots &\vdots &\ddots &\vdots \\\alpha _{l}(X_{1})&\alpha _{l}(X_{2})&\dots &\alpha _{l}(X_{l})\end{bmatrix}}$

### Projection and rejection

    $(-1)^{k}\iota _{X}{\star }\alpha ={\star }(X^{\flat }\wedge \alpha )$ ( interior product $\iota _{X}{\star }$ dual to wedge $X^{\flat }\wedge$ )
    $(\iota _{X}\alpha )\wedge {\star }\beta =\alpha \wedge {\star }(X^{\flat }\wedge \beta )$ for $\alpha \in \Omega ^{k+1}(M),\beta \in \Omega ^{k}(M)$
If $|X|=1,\ \alpha \in \Omega ^{k}(M)$, then
* $\iota _{X}\circ (X^{\flat }\wedge ):\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)$ is the projection of $\alpha$ onto the orthogonal complement of $X$.
* $(X^{\flat }\wedge )\circ \iota _{X}:\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)$ is the rejection of $\alpha$, the remainder of the projection.
* thus $\iota _{X}\circ (X^{\flat }\wedge )+(X^{\flat }\wedge )\circ \iota _{X}={\text{id}}$ ( projection–rejection decomposition )
Given the boundary $\partial M$ with unit normal vector $N$
* $\mathbf {t} :=\iota _{N}\circ (N^{\flat }\wedge )$ extracts the tangential component of the boundary.
* $\mathbf {n} :=({\text{id}}-\mathbf {t} )$ extracts the normal component of the boundary.

### Sum expressions

    $(d\alpha )(X_{0},\ldots ,X_{k})=\sum _{0\leq j\leq k}(-1)^{j}d(\alpha (X_{0},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}))(X_{j})+\sum _{0\leq i<j\leq k}(-1)^{i+j}\alpha ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k})$
    $(d\alpha )(X_{1},\ldots ,X_{k})=\sum _{i=1}^{k}(-1)^{i+1}(\nabla _{X_{i}}\alpha )(X_{1},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})$
    $(\delta \alpha )(X_{1},\ldots ,X_{k-1})=-\sum _{i=1}^{n}(\iota _{E_{i}}(\nabla _{E_{i}}\alpha ))(X_{1},\ldots ,{\hat {X}}_{i},\ldots ,X_{k})$ given a positively oriented orthonormal frame $E_{1},\ldots ,E_{n}$.
    $({\mathcal {L}}_{Y}\alpha )(X_{1},\ldots ,X_{k})=(\nabla _{Y}\alpha )(X_{1},\ldots ,X_{k})-\sum _{i=1}^{k}\alpha (X_{1},\ldots ,\nabla _{X_{i}}Y,\ldots ,X_{k})$

### Hodge decomposition

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

### Poincaré lemma

If a boundaryless manifold $M$ has trivial cohomology $H^{k}(M)=\{0\}$, then any closed $\omega \in \Omega ^{k}(M)$ 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 $\nabla =\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)$ differential operator $\mathbb {R} ^{3}$
    $\iota _{X}\alpha =g(X,\alpha ^{\sharp })=X\cdot \alpha ^{\sharp }$ for $\alpha \in \Omega ^{1}(M)$.
    $\mathbf {det} (X,Y,Z)=\langle X,Y\times Z\rangle =\langle X\times Y,Z\rangle$ ( scalar triple product )
    $X\times Y=({\star }(X^{\flat }\wedge Y^{\flat }))^{\sharp }$ ( cross product )
    $\iota _{X}\alpha =-(X\times A)^{\flat }$ if $\alpha \in \Omega ^{2}(M),\ A=({\star }\alpha )^{\sharp }$
    $X\cdot Y={\star }(X^{\flat }\wedge {\star }Y^{\flat })$ ( scalar product )
    $\nabla f=(df)^{\sharp }$ ( gradient )
    $X\cdot \nabla f=df(X)$ ( directional derivative )
    $\nabla \cdot X={\star }d{\star }X^{\flat }=-\delta X^{\flat }$ ( divergence )
    $\nabla \times X=({\star }dX^{\flat })^{\sharp }$ ( curl )
    $\langle X,N\rangle \sigma ={\star }X^{\flat }$ where $N$ is the unit normal vector of $\partial M$ and $\sigma =\iota _{N}\mathbf {det}$ is the area form on $\partial M$.
    $\int _{\Sigma }d{\star }X^{\flat }=\int _{\partial \Sigma }{\star }X^{\flat }=\int _{\partial \Sigma }\langle X,N\rangle \sigma$ ( divergence theorem )

### Lie derivatives

    ${\mathcal {L}}_{X}f=X\cdot \nabla f$ ( $0$-forms )
    ${\mathcal {L}}_{X}\alpha =(\nabla _{X}\alpha ^{\sharp })^{\flat }+g(\alpha ^{\sharp },\nabla X)$ ( $1$-forms )
    ${\star }{\mathcal {L}}_{X}\beta =\left(\nabla _{X}B-\nabla _{B}X+({\text{div}}X)B\right)^{\flat }$ if $B=({\star }\beta )^{\sharp }$ ( $2$-forms on $3$-manifolds )
    ${\star }{\mathcal {L}}_{X}\rho =dq(X)+({\text{div}}X)q$ if $\rho ={\star }q\in \Omega ^{0}(M)$ ( $n$-forms )
    ${\mathcal {L}}_{X}(\mathbf {det} )=({\text{div}}(X))\mathbf {det}$