Exterior calculus identities

This article summarizes important identities in exterior calculus created from the sources , , , , .

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 two points on the manifolds.

$TM$ is the tangent bundle of the smooth manifold $M$.

$T_{p}M$, $T_{q}N$ denote the tangent spaces of $M$, $N$ at the points $p$, $q$, respectively.

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 $X|_{p},Y|_{p},Z|_{p}\in T_{p}M$.

Given an inner product $g(\cdot ,\cdot )_{p}$ on each $T_{p}M$, the manifold becomes a Riemannian manifold.

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 $\Sigma \subset M$.

k-forms

$k$-forms 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 $\alpha \in \Omega ^{k}(M)$, $\beta \in \Omega ^{l}(M)$, $\gamma \in \Omega ^{m}(M)$. In this article we focus on $\mathbb {R}$ valued forms.

$0$-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 omit 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

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

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

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

Exterior derivative

The exterior derivative $d:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M)$ is defined for all $0\leq k\leq n$. For a $0$-form $f\in \Omega ^{k}(M)$ we have $df\in \Omega ^{1}(M)$ as the directional derivative $1$-form. i.e. in the direction $X\in T_{p}M$ we have $df(X)=Xf$.

For $0<k\leq n$ it can be written as

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

The recursiveness of the above definition is finite, since $d$ for $0$-forms is defined separately.

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

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

Musical isomorphisms

Given a section $A\in \Gamma (TM)$ there exists a $1$-form $A^{\flat }\in \Omega ^{1}(M)$ such that on each $p\in M$

$\forall X\in T_{p}M\Rightarrow A^{\flat }(X)=g(A,X)_{p}.$

We call this mapping the flat operator $\flat$.

Given a $1$-form $\alpha \in \Omega ^{1}(M)$ there exists a section $\alpha ^{\sharp }\in \Gamma (TM)$ such that on each $p\in M$

$\forall X\in T_{p}M\Rightarrow g(\alpha |_{p}^{\sharp },X)_{p}=\alpha |_{p}(X).$

We call this mapping the sharp operator $\sharp$.

$\sharp$ and $\flat$ constitute the musical isomorphisms.

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 $X$. If $\alpha \in \Omega ^{k+1}(M)$ and $X_{i}\in \Gamma (TM)$ then

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

Hodge star

The Hodge star operator ${\star }:\Omega ^{k}(M)\rightarrow \Omega ^{n-k}(M)$ is defined as such that it maps $k$-forms $\alpha \in \Omega ^{k}(M)$ to their dual $(n-k)$-form $({\star }\alpha )\ \Omega ^{n-k}(M)$.

For example, if $(X_{1},\ldots ,X_{n})$ is a positively oriented frame for $TM$ according to the given metric $g$, then

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

We omit to write the dimension $k$ or inversion symbol $-1$ with the Hodge star operator as it is evident in the context.

We call $s={\text{sign}}(g)$ the signature of the metric $g$. For example in Minkowski space $s=-1$ and in Riemannian manifolds $s=1$.

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

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

Oriented manifold

A $n$ dimensional oriented manifold $M$ is a manifold that can be equipped with a choice of a non-zero $n$-form $\mu \in \Omega ^{n}(M)$.

Volume form

On a orientable manifold $M$ the canonical choice of a volume form given a metric $g$ is $\mathbf {det} :={\sqrt {\det g}}\;dX_{1}^{\flat }\wedge \ldots \wedge dX_{n}^{\flat }$ for any positively oriented basis $dX_{1},\ldots ,dX_{n}$.

Area form

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

Inner product for k-forms

The inner product 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 $L^{2}$-inner product for the space $k$-forms $\Omega ^{k}(M)$ is defined by

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

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 map $\phi _{t}$ 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 $\exists \beta \in \Omega ^{k-1}:\ \alpha =d\beta$
• coclosed if $\delta \alpha =0$
• coexact if $\exists \beta \in \Omega ^{k+1}:\ \alpha =\delta \beta$
• 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

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

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.

$[\alpha ]=[\beta ]\Leftrightarrow \exists \ \eta \in \Omega ^{k-1}(M)\ :\ \alpha -\beta =d\eta$

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

Dirichlet energy

Given $\alpha \in \Omega ^{k}(M)$

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

Properties

Exterior derivative properties

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

$d\circ d=0$ ( Nilpotent )

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

$df(X)=Xf$ if $f\in \Omega ^{0}(M),\ X\in \Gamma (TM)$ ( Directional derivative )

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

Exterior product properties

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

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

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

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

$\alpha \wedge \alpha =0$ when $\operatorname {rank} \alpha \leq 1$. The rank of a $k$-form $\alpha$ is defined as the minimum number of terms, each consisting of the exterior product of $k$ $1$-forms, that can 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

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

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

Inner product properties

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

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

$\iota _{X}(\alpha \wedge \beta )=(\iota _{X}\alpha )\wedge \beta +(-1)^{k}\alpha \wedge (\iota _{X}\beta )=0$ 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 )

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

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

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

Co-differential operator properties

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

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

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

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

$\iota _{X}({\star }\mathbf {1} )={\star }X^{\flat }$ if $f\in \Omega ^{0}(M)$

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

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

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

$X^{\flat }\wedge {\star }Y^{\flat }=g(X,Y)({\star }\mathbf {1} )$ ( Inner product )

$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$ ( 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

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

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

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

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

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 $M$ has only one cohomology class $H^{k}(M)=\{0\}$ and no boundary $\partial M=\emptyset$, then for any closed $\omega \in \Omega ^{k}(M)\Rightarrow \exists \ \alpha \in \Omega ^{k-1}$ such that

$\omega =d\alpha$

Relations to vector calculus

See also: Vector calculus identities

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 $\alpha \in \Omega ^{1}(M)$.

${\text{det}}(X,Y,Z)=\langle X,Y\times Z\rangle =\langle X\times Y,Z\rangle$ ( Cross product )

• ${\star }(\alpha \wedge \beta )=\alpha ^{\sharp }\times \beta ^{\sharp }$

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

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

$\nabla f=(df)^{\sharp }$ ( Gradient $1$-form )

$X\cdot \nabla f=df(X)$ ( Directional derivative )

$\nabla \cdot X={\star }d{\star }X^{\flat }=\delta X^{\flat }$ ( Divergence )

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

$\langle X,N\rangle ={\star }X^{\flat }$ if $N$ is the unit normal vector on $\partial M$

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

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