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 $$d_0f\in\Omega^1(M)$$ 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 $$ 0 < k\leq n$$,


 * $$ (d_k\omega)(X_0,\ldots,X_k)=\sum_{0\leq j\leq k}(-1)^jd_{0}(\omega(X_0,\ldots,\hat{X}_j,\ldots,X_k))(X_j) + \sum_{0\leq i < j\leq k}(-1)^{i+j}\omega([X_i,X_j],X_0,\ldots,\hat{X}_i,\ldots,\hat{X}_j,\ldots,X_k) .$$

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 $$d\phi|_p:T_pM\rightarrow T_{\phi(p)}N$$ defines a tangent map from $$M$$ to $$N$$. It is defined through curves $$\gamma$$ on $$M$$ with derivative $$\gamma'(0)=X\in T_pM$$ 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 $$\Sigma\subset M$$


 * $$ \int_{\Sigma} \phi^*\alpha = \int_{\phi(\Sigma)} \alpha .$$

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 $$\Delta = -(d\delta + \delta d)$$.

Definitions on Ωk(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 $$

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 \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^{\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)^kd{\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 \ \exist \ 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)^jd(\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 $$\partial M =\empty$$, $$\omega\in\Omega^k(M) \Rightarrow \exists \alpha\in\Omega^{k-1}, \ \beta\in\Omega^{k+1}, \ \gamma\in\Omega^k(M), \ d\gamma=0, \ \delta\gamma = 0$$ 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.

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} d X^{\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_XB - \nabla_BX + (\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} $$