Английская Википедия:Exterior calculus identities
Шаблон:Short description Шаблон:Context 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
<math>M</math>, <math>N</math> are <math>n</math>-dimensional smooth manifolds, where <math> n\in \mathbb{N} </math>. That is, differentiable manifolds that can be differentiated enough times for the purposes on this page.
<math> p \in M </math>, <math> q \in N </math> denote one point on each of the manifolds.
The boundary of a manifold <math> M </math> is a manifold <math> \partial M </math>, which has dimension <math> n - 1 </math>. An orientation on <math> M </math> induces an orientation on <math> \partial M </math>.
We usually denote a submanifold by <math>\Sigma \subset M</math>.
Tangent and cotangent bundles
<math>TM</math>, <math>T^{*}M</math> denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold <math>M</math>.
<math> T_p M </math>, <math> T_q N </math> denote the tangent spaces of <math>M</math>, <math>N</math> at the points <math>p</math>, <math>q</math>, respectively. <math> T^{*}_p M </math> denotes the cotangent space of <math>M</math> at the point <math>p</math>.
Sections of the tangent bundles, also known as vector fields, are typically denoted as <math>X, Y, Z \in \Gamma(TM)</math> such that at a point <math> p \in M </math> we have <math> X|_p, Y|_p, Z|_p \in T_p M </math>. Sections of the cotangent bundle, also known as differential 1-forms (or covector fields), are typically denoted as <math>\alpha, \beta \in \Gamma(T^{*}M)</math> such that at a point <math> p \in M </math> we have <math> \alpha|_p, \beta|_p \in T^{*}_p M </math>. An alternative notation for <math>\Gamma(T^{*}M)</math> is <math>\Omega^1(M)</math>.
Differential k-forms
Differential <math>k</math>-forms, which we refer to simply as <math>k</math>-forms here, are differential forms defined on <math>TM</math>. We denote the set of all <math>k</math>-forms as <math>\Omega^k(M)</math>. For <math> 0\leq k,\ l,\ m\leq n </math> we usually write <math>\alpha\in\Omega^k(M)</math>, <math>\beta\in\Omega^l(M)</math>, <math>\gamma\in\Omega^m(M)</math>.
<math>0</math>-forms <math>f\in\Omega^0(M)</math> are just scalar functions <math>C^{\infty}(M)</math> on <math>M</math>. <math>\mathbf{1}\in\Omega^0(M)</math> denotes the constant <math>0</math>-form equal to <math>1</math> everywhere.
Omitted elements of a sequence
When we are given <math>(k+1)</math> inputs <math>X_0,\ldots,X_k</math> and a <math>k</math>-form <math>\alpha\in\Omega^k(M)</math> we denote omission of the <math>i</math>th entry by writing
- <math>\alpha(X_0,\ldots,\hat{X}_i,\ldots,X_k):=\alpha(X_0,\ldots,X_{i-1},X_{i+1},\ldots,X_k) .</math>
Exterior product
The exterior product is also known as the wedge product. It is denoted by <math> \wedge : \Omega^k(M) \times \Omega^l(M) \rightarrow \Omega^{k+l}(M)</math>. The exterior product of a <math>k</math>-form <math>\alpha\in\Omega^k(M)</math> and an <math>l</math>-form <math>\beta\in\Omega^l(M)</math> produce a <math>(k+l)</math>-form <math>\alpha\wedge\beta \in\Omega^{k+l}(M)</math>. It can be written using the set <math>S(k,k+l)</math> of all permutations <math>\sigma</math> of <math>\{1,\ldots,n\}</math> such that <math>\sigma(1)<\ldots <\sigma(k), \ \sigma(k+1)<\ldots <\sigma(k+l) </math> as
- <math>(\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)}) .</math>
Directional derivative
The directional derivative of a 0-form <math>f\in\Omega^0(M)</math> along a section <math>X\in\Gamma(TM)</math> is a 0-form denoted <math>\partial_X f .</math>
Exterior derivative
The exterior derivative <math>d_k : \Omega^k(M) \rightarrow \Omega^{k+1}(M) </math> is defined for all <math> 0 \leq k\leq n</math>. We generally omit the subscript when it is clear from the context.
For a <math>0</math>-form <math>f\in\Omega^0(M)</math> we have <math>d_0f\in\Omega^1(M)</math> as the <math>1</math>-form that gives the directional derivative, i.e., for the section <math>X\in \Gamma(TM)</math> we have <math>(d_0f)(X) = \partial_X f</math>, the directional derivative of <math>f</math> along <math>X</math>.[6]
For <math> 0 < k\leq n</math>,[6]
- <math> (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) .</math>
Lie bracket
The Lie bracket of sections <math>X,Y \in \Gamma(TM)</math> is defined as the unique section <math>[X,Y] \in \Gamma(TM)</math> that satisfies
- <math>
\forall f\in\Omega^0(M) \Rightarrow \partial_{[X,Y]}f = \partial_X \partial_Y f - \partial_Y \partial_X f . </math>
Tangent maps
If <math> \phi : M \rightarrow N </math> is a smooth map, then <math>d\phi|_p:T_pM\rightarrow T_{\phi(p)}N</math> defines a tangent map from <math>M</math> to <math>N</math>. It is defined through curves <math>\gamma</math> on <math>M</math> with derivative <math>\gamma'(0)=X\in T_pM</math> such that
- <math>d\phi(X):=(\phi\circ\gamma)' .</math>
Note that <math>\phi</math> is a <math>0</math>-form with values in <math>N</math>.
Pull-back
If <math> \phi : M \rightarrow N </math> is a smooth map, then the pull-back of a <math>k</math>-form <math> \alpha\in \Omega^k(N) </math> is defined such that for any <math>k</math>-dimensional submanifold <math>\Sigma\subset M</math>
- <math> \int_{\Sigma} \phi^*\alpha = \int_{\phi(\Sigma)} \alpha .</math>
The pull-back can also be expressed as
- <math>(\phi^*\alpha)(X_1,\ldots,X_k)=\alpha(d\phi(X_1),\ldots,d\phi(X_k)) .</math>
Interior product
Also known as the interior derivative, the interior product given a section <math> Y\in \Gamma(TM) </math> is a map <math>\iota_Y:\Omega^{k+1}(M) \rightarrow \Omega^k(M)</math> that effectively substitutes the first input of a <math>(k+1)</math>-form with <math>Y</math>. If <math>\alpha\in\Omega^{k+1}(M)</math> and <math>X_i\in \Gamma(TM)</math> then
- <math> (\iota_Y\alpha)(X_1,\ldots,X_k) = \alpha(Y,X_1,\ldots,X_k) .</math>
Metric tensor
Given a nondegenerate bilinear form <math> g_p( \cdot , \cdot ) </math> on each <math> T_p M </math> that is continuous on <math>M</math>, the manifold becomes a pseudo-Riemannian manifold. We denote the metric tensor <math>g</math>, defined pointwise by <math> g( X , Y )|_p = g_p( X|_p , Y|_p ) </math>. We call <math>s=\operatorname{sign}(g)</math> the signature of the metric. A Riemannian manifold has <math>s=1</math>, whereas Minkowski space has <math>s=-1</math>.
Musical isomorphisms
The metric tensor <math>g(\cdot,\cdot)</math> induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat <math>\flat</math> and sharp <math>\sharp</math>. A section <math> A \in \Gamma(TM)</math> corresponds to the unique one-form <math>A^{\flat}\in\Omega^1(M)</math> such that for all sections <math>X \in \Gamma(TM)</math>, we have:
- <math> A^{\flat}(X) = g(A,X) .</math>
A one-form <math>\alpha\in\Omega^1(M)</math> corresponds to the unique vector field <math> \alpha^{\sharp}\in \Gamma(TM)</math> such that for all <math>X \in \Gamma(TM)</math>, we have:
- <math> \alpha(X) = g(\alpha^\sharp,X) .</math>
These mappings extend via multilinearity to mappings from <math>k</math>-vector fields to <math>k</math>-forms and <math>k</math>-forms to <math>k</math>-vector fields through
- <math> (A_1 \wedge A_2 \wedge \cdots \wedge A_k)^{\flat} = A_1^{\flat} \wedge A_2^{\flat} \wedge \cdots \wedge A_k^{\flat}</math>
- <math> (\alpha_1 \wedge \alpha_2 \wedge \cdots \wedge \alpha_k)^{\sharp} = \alpha_1^{\sharp} \wedge \alpha_2^{\sharp} \wedge \cdots \wedge \alpha_k^{\sharp}.</math>
Hodge star
For an n-manifold M, the Hodge star operator <math>{\star}:\Omega^k(M)\rightarrow\Omega^{n-k}(M)</math> is a duality mapping taking a <math>k</math>-form <math>\alpha \in \Omega^k(M)</math> to an <math>(n{-}k)</math>-form <math>({\star}\alpha) \in \Omega^{n-k}(M)</math>.
It can be defined in terms of an oriented frame <math>(X_1,\ldots,X_n)</math> for <math>TM</math>, orthonormal with respect to the given metric tensor <math>g</math>:
- <math>
({\star}\alpha)(X_1,\ldots,X_{n-k})=\alpha(X_{n-k+1},\ldots,X_n) . </math>
Co-differential operator
The co-differential operator <math>\delta:\Omega^k(M)\rightarrow\Omega^{k-1}(M)</math> on an <math>n</math> dimensional manifold <math>M</math> is defined by
- <math>\delta := (-1)^{k} {\star}^{-1} d {\star} = (-1)^{nk+n+1}{\star} d {\star} .</math>
The Hodge–Dirac operator, <math>d+\delta</math>, is a Dirac operator studied in Clifford analysis.
Oriented manifold
An <math>n</math>-dimensional orientable manifold Шаблон:Mvar is a manifold that can be equipped with a choice of an Шаблон:Mvar-form <math>\mu\in\Omega^n(M)</math> that is continuous and nonzero everywhere on Шаблон:Mvar.
Volume form
On an orientable manifold <math>M</math> the canonical choice of a volume form given a metric tensor <math>g</math> and an orientation is <math>\mathbf{det}:=\sqrt{|\det g|}\;dX_1^{\flat}\wedge\ldots\wedge dX_n^{\flat}</math> for any basis <math>dX_1,\ldots, dX_n</math> ordered to match the orientation.
Area form
Given a volume form <math>\mathbf{det}</math> and a unit normal vector <math>N</math> we can also define an area form <math>\sigma:=\iota_N\textbf{det}</math> on the Шаблон:Nowrap
Bilinear form on k-forms
A generalization of the metric tensor, the symmetric bilinear form between two <math>k</math>-forms <math>\alpha,\beta\in\Omega^k(M)</math>, is defined pointwise on <math>M</math> by
- <math>
\langle\alpha,\beta\rangle|_p := {\star}(\alpha\wedge {\star}\beta )|_p . </math>
The <math>L^2</math>-bilinear form for the space of <math>k</math>-forms <math>\Omega^k(M)</math> is defined by
- <math>
\langle\!\langle\alpha,\beta\rangle\!\rangle:= \int_M\alpha\wedge {\star}\beta . </math>
In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).
Lie derivative
We define the Lie derivative <math>\mathcal{L}:\Omega^k(M)\rightarrow\Omega^k(M)</math> through Cartan's magic formula for a given section <math>X\in \Gamma(TM)</math> as
- <math>
\mathcal{L}_X = d \circ \iota_X + \iota_X \circ d . </math>
It describes the change of a <math>k</math>-form along a flow <math>\phi_t</math> associated to the section <math>X</math>.
Laplace–Beltrami operator
The Laplacian <math>\Delta:\Omega^k(M) \rightarrow \Omega^k(M)</math> is defined as <math>\Delta = -(d\delta + \delta d)</math>.
Important definitions
Definitions on Ωk(M)
<math>\alpha\in\Omega^k(M)</math> is called...
- closed if <math>d\alpha=0</math>
- exact if <math> \alpha = d\beta</math> for some <math>\beta\in\Omega^{k-1}</math>
- coclosed if <math>\delta\alpha=0</math>
- coexact if <math> \alpha = \delta\beta</math> for some <math>\beta\in\Omega^{k+1}</math>
- harmonic if closed and coclosed
Cohomology
The <math>k</math>-th cohomology of a manifold <math>M</math> and its exterior derivative operators <math>d_0,\ldots,d_{n-1}</math> is given by
- <math>
H^k(M):=\frac{\text{ker}(d_{k})}{\text{im}(d_{k-1})} </math>
Two closed <math>k</math>-forms <math>\alpha,\beta\in\Omega^k(M)</math> are in the same cohomology class if their difference is an exact form i.e.
- <math>
[\alpha]=[\beta] \ \ \Longleftrightarrow\ \ \alpha{-}\beta = d\eta \ \text{ for some } \eta\in\Omega^{k-1}(M) </math>
A closed surface of genus <math>g</math> will have <math>2g</math> generators which are harmonic.
Dirichlet energy
Given <math>\alpha\in\Omega^k(M)</math>, its Dirichlet energy is
- <math>
\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 </math>
Properties
Exterior derivative properties
- <math>
\int_{\Sigma} d\alpha = \int_{\partial\Sigma} \alpha </math> ( Stokes' theorem )
- <math>
d \circ d = 0 </math> ( cochain complex )
- <math>
d(\alpha \wedge \beta ) = d\alpha\wedge \beta +(-1)^k\alpha\wedge d\beta </math> for <math> \alpha\in\Omega^k(M), \ \beta\in\Omega^l(M) </math> ( Leibniz rule )
- <math>
df(X) = \partial_X f </math> for <math> f\in\Omega^0(M), \ X\in \Gamma(TM) </math> ( directional derivative )
- <math>
d\alpha = 0 </math> for <math>\alpha \in \Omega^n(M), \ \text{dim}(M)=n </math>
Exterior product properties
- <math>
\alpha \wedge \beta = (-1)^{kl}\beta \wedge \alpha </math> for <math> \alpha\in\Omega^k(M), \ \beta\in\Omega^l(M) </math> ( alternating )
- <math>
(\alpha \wedge \beta)\wedge\gamma = \alpha \wedge (\beta\wedge\gamma) </math> ( associativity )
- <math>
(\lambda\alpha) \wedge \beta = \lambda (\alpha \wedge \beta) </math> for <math>\lambda\in\mathbb{R}</math> ( compatibility of scalar multiplication )
- <math>
\alpha \wedge ( \beta_1 + \beta_2 ) = \alpha \wedge \beta_1 + \alpha \wedge \beta_2 </math> ( distributivity over addition )
- <math>
\alpha \wedge \alpha = 0 </math> for <math> \alpha\in\Omega^k(M) </math> when <math>k</math> is odd or <math>\operatorname{rank} \alpha \le 1 </math>. The rank of a <math>k</math>-form <math>\alpha</math> means the minimum number of monomial terms (exterior products of one-forms) that must be summed to produce <math>\alpha</math>.
Pull-back properties
- <math>
d(\phi^*\alpha) = \phi^*(d\alpha) </math> ( commutative with <math>d</math> )
- <math>
\phi^*(\alpha\wedge\beta) = (\phi^*\alpha)\wedge(\phi^*\beta) </math> ( distributes over <math>\wedge</math> )
- <math>
(\phi_1\circ\phi_2)^* = \phi_2^*\phi_1^* </math> ( contravariant )
- <math>
\phi^*f=f\circ\phi </math> for <math>f\in\Omega^0(N)</math> ( function composition )
Musical isomorphism properties
- <math>
(X^{\flat})^{\sharp}=X </math>
- <math>
(\alpha^{\sharp})^{\flat}=\alpha </math>
Interior product properties
- <math>
\iota_X \circ \iota_X = 0 </math> ( nilpotent )
- <math>
\iota_X \circ \iota_Y = - \iota_Y \circ \iota_X </math>
- <math>
\iota_X (\alpha \wedge \beta ) = (\iota_X\alpha)\wedge\beta + (-1)^k\alpha\wedge(\iota_X \beta ) </math> for <math>\alpha\in\Omega^k(M), \ \beta\in\Omega^l(M)</math> ( Leibniz rule )
- <math>
\iota_X\alpha = \alpha(X) </math> for <math>\alpha\in\Omega^1(M)</math>
- <math>
\iota_X f = 0 </math> for <math>f \in \Omega^0(M)</math>
- <math>
\iota_X(f\alpha) = f \iota_X\alpha </math> for <math>f \in \Omega^0(M)</math>
Hodge star properties
- <math>
{\star}(\lambda_1\alpha + \lambda_2\beta) = \lambda_1({\star}\alpha) + \lambda_2({\star}\beta) </math> for <math>\lambda_1,\lambda_2\in\mathbb{R}</math> ( linearity )
- <math>
{\star}{\star}\alpha = s(-1)^{k(n-k)}\alpha </math> for <math>\alpha\in \Omega^k(M)</math>, <math>n=\dim(M)</math>, and <math>s = \operatorname{sign}(g)</math> the sign of the metric
- <math>
{\star}^{(-1)} = s(-1)^{k(n-k)}{\star} </math> ( inversion )
- <math>
{\star}(f\alpha)=f({\star}\alpha) </math> for <math>f\in\Omega^0(M)</math> ( commutative with <math>0</math>-forms )
- <math>
\langle\!\langle\alpha,\alpha\rangle\!\rangle = \langle\!\langle{\star}\alpha,{\star}\alpha\rangle\!\rangle </math> for <math>\alpha\in\Omega^1(M)</math> ( Hodge star preserves <math>1</math>-form norm )
- <math>
{\star} \mathbf{1} = \mathbf{det} </math> ( Hodge dual of constant function 1 is the volume form )
Co-differential operator properties
- <math>
\delta\circ\delta = 0 </math> ( nilpotent )
- <math>
{\star}\delta=(-1)^kd{\star} </math> and <math>{\star} d = (-1)^{k+1}\delta{\star}</math> ( Hodge adjoint to <math>d</math> )
- <math>
\langle\!\langle d\alpha,\beta\rangle\!\rangle = \langle\!\langle \alpha,\delta\beta\rangle\!\rangle </math> if <math>\partial M=0</math> ( <math>\delta</math> adjoint to <math>d</math> )
- In general, <math>\int_M d\alpha \wedge \star \beta = \int_{\partial M} \alpha \wedge \star \beta + \int_M \alpha\wedge\star\delta\beta </math>
- <math>
\delta f = 0 </math> for <math>f \in \Omega^0(M)</math>
Lie derivative properties
- <math>
d\circ\mathcal{L}_X = \mathcal{L}_X\circ d </math> ( commutative with <math>d</math> )
- <math>
\iota_X \circ\mathcal{L}_X = \mathcal{L}_X\circ \iota_X </math> ( commutative with <math>\iota_X</math> )
- <math>
\mathcal{L}_X(\iota_Y\alpha) = \iota_{[X,Y]}\alpha + \iota_Y\mathcal{L}_X\alpha </math>
- <math>
\mathcal{L}_X(\alpha\wedge\beta) = (\mathcal{L}_X\alpha)\wedge\beta + \alpha\wedge(\mathcal{L}_X\beta) </math> ( Leibniz rule )
Exterior calculus identities
- <math>
\iota_X({\star}\mathbf{1}) = {\star} X^{\flat} </math>
- <math>
\iota_X({\star}\alpha) = (-1)^k{\star}(X^{\flat}\wedge\alpha) </math> if <math>\alpha\in\Omega^k(M)</math>
- <math>
\iota_X(\phi^*\alpha)=\phi^*(\iota_{d\phi(X)}\alpha) </math>
- <math>
\nu,\mu\in\Omega^n(M), \mu \text{ non-zero } \ \Rightarrow \ \exist \ f\in\Omega^0(M): \ \nu=f\mu </math>
- <math>
X^{\flat}\wedge{\star} Y^{\flat} = g(X,Y)( {\star} \mathbf{1}) </math> ( bilinear form )
- <math>
[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]] = 0 </math> ( Jacobi identity )
Dimensions
If <math>n=\dim M</math>
- <math>
\dim\Omega^k(M) = \binom{n}{k} </math> for <math>0\leq k\leq n</math>
- <math>
\dim\Omega^k(M) = 0 </math> for <math>k < 0, \ k > n</math>
If <math>X_1,\ldots,X_n\in \Gamma(TM)</math> is a basis, then a basis of <math>\Omega^k(M)</math> is
- <math>
\{X_{\sigma(1)}^{\flat}\wedge\ldots\wedge X_{\sigma(k)}^{\flat} \ : \ \sigma\in S(k,n)\} </math>
Exterior products
Let <math>\alpha, \beta, \gamma,\alpha_i\in \Omega^1(M)</math> and <math>X,Y,Z,X_i</math> be vector fields.
- <math>
\alpha(X) = \det \begin{bmatrix}
\alpha(X) \\
\end{bmatrix}
</math>
- <math>
(\alpha\wedge\beta)(X,Y) = \det \begin{bmatrix}
\alpha(X) & \alpha(Y) \\
\beta(X) & \beta(Y) \\
\end{bmatrix}
</math>
- <math>
(\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}
</math>
- <math>
(\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}
</math>
Projection and rejection
- <math>
(-1)^k\iota_X{\star}\alpha = {\star}(X^{\flat}\wedge\alpha) </math> ( interior product <math>\iota_X{\star}</math> dual to wedge <math>X^{\flat}\wedge</math> )
- <math>
(\iota_X\alpha)\wedge{\star}\beta =\alpha\wedge{\star}(X^{\flat}\wedge\beta) </math> for <math>\alpha\in\Omega^{k+1}(M),\beta\in\Omega^k(M)</math>
If <math>|X|=1, \ \alpha\in\Omega^k(M)</math>, then
- <math>\iota_X\circ (X^{\flat}\wedge ):\Omega^k(M)\rightarrow\Omega^k(M)</math> is the projection of <math>\alpha</math> onto the orthogonal complement of <math>X</math>.
- <math>(X^{\flat}\wedge )\circ \iota_X:\Omega^k(M)\rightarrow\Omega^k(M)</math> is the rejection of <math>\alpha</math>, the remainder of the projection.
- thus <math> \iota_X \circ (X^{\flat}\wedge ) + (X^{\flat}\wedge)\circ\iota_X = \text{id} </math> ( projection–rejection decomposition )
Given the boundary <math>\partial M</math> with unit normal vector <math>N</math>
- <math>\mathbf{t}:=\iota_N\circ (N^{\flat}\wedge )</math> extracts the tangential component of the boundary.
- <math>\mathbf{n}:=(\text{id}-\mathbf{t})</math> extracts the normal component of the boundary.
Sum expressions
- <math>
(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) </math>
- <math>
(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) </math>
- <math>
(\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) </math> given a positively oriented orthonormal frame <math>E_1,\ldots,E_n</math>.
- <math>
(\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) </math>
Hodge decomposition
If <math>\partial M =\empty</math>, <math>\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</math> such thatШаблон:Citation needed
- <math>
\omega = d\alpha + \delta\beta + \gamma </math>
Poincaré lemma
If a boundaryless manifold <math>M</math> has trivial cohomology <math>H^k(M)=\{0\}</math>, then any closed <math>\omega\in\Omega^k(M)</math> is exact. This is the case if M is contractible.
Relations to vector calculus
Identities in Euclidean 3-space
Let Euclidean metric <math>g(X,Y):=\langle X,Y\rangle = X\cdot Y</math>.
We use <math> \nabla = \left( {\partial \over \partial x}, {\partial \over \partial y}, {\partial \over \partial z} \right) </math> differential operator <math>\mathbb{R}^3</math>
- <math>
\iota_X\alpha = g(X,\alpha^{\sharp}) = X\cdot \alpha^{\sharp} </math> for <math>\alpha\in\Omega^1(M)</math>.
- <math>
\mathbf{det}(X,Y,Z)=\langle X,Y\times Z\rangle = \langle X\times Y,Z\rangle </math> ( scalar triple product )
- <math>
X\times Y = ({\star}(X^{\flat}\wedge Y^{\flat}))^{\sharp} </math> ( cross product )
- <math>
\iota_X\alpha=-(X\times A)^{\flat} </math> if <math>\alpha\in\Omega^2(M),\ A=({\star}\alpha)^{\sharp}</math>
- <math>
X\cdot Y = {\star}(X^{\flat}\wedge {\star} Y^{\flat}) </math> ( scalar product )
- <math>
\nabla f=(df)^{\sharp} </math> ( gradient )
- <math>
X\cdot\nabla f=df(X) </math> ( directional derivative )
- <math>
\nabla\cdot X = {\star} d {\star} X^{\flat} = -\delta X^{\flat} </math> ( divergence )
- <math>
\nabla\times X = ({\star} d X^{\flat})^{\sharp} </math> ( curl )
- <math>
\langle X,N\rangle\sigma = {\star} X^\flat </math> where <math>N</math> is the unit normal vector of <math>\partial M</math> and <math>\sigma=\iota_{N}\mathbf{det}</math> is the area form on <math>\partial M</math>.
- <math>
\int_{\Sigma} d{\star} X^{\flat} = \int_{\partial\Sigma}{\star} X^{\flat} = \int_{\partial\Sigma}\langle X,N\rangle\sigma </math> ( divergence theorem )
Lie derivatives
- <math>
\mathcal{L}_X f =X\cdot \nabla f </math> ( <math>0</math>-forms )
- <math>
\mathcal{L}_X \alpha = (\nabla_X\alpha^{\sharp})^{\flat} +g(\alpha^{\sharp},\nabla X) </math> ( <math>1</math>-forms )
- <math>
{\star}\mathcal{L}_X\beta = \left( \nabla_XB - \nabla_BX + (\text{div}X)B \right)^{\flat} </math> if <math>B=({\star}\beta)^{\sharp}</math> ( <math>2</math>-forms on <math>3</math>-manifolds )
- <math>
{\star}\mathcal{L}_X\rho = dq(X)+(\text{div}X)q </math> if <math>\rho={\star} q \in \Omega^0(M)</math> ( <math>n</math>-forms )
- <math>
\mathcal{L}_X(\mathbf{det})=(\text{div}(X))\mathbf{det} </math>
References
