This article summarizes several identities in exterior calculus, a mathematical notation used in differential geometry.[1][2][3][4][5]
Notation
The following summarizes short definitions and notations that are used in this article.
Manifold
,
are
-dimensional smooth manifolds, where
. That is, differentiable manifolds that can be differentiated enough times for the purposes on this page.
,
denote one point on each of the manifolds.
The boundary of a manifold
is a manifold
, which has dimension
. An orientation on
induces an orientation on
.
We usually denote a submanifold by
.
Tangent and cotangent bundles
,
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold
.
,
denote the tangent spaces of
,
at the points
,
, respectively.
denotes the cotangent space of
at the point
.
Sections of the tangent bundles, also known as vector fields, are typically denoted as
such that at a point
we have
. Sections of the cotangent bundle, also known as differential 1-forms (or covector fields), are typically denoted as
such that at a point
we have
. An alternative notation for
is
.
Differential
-forms, which we refer to simply as
-forms here, are differential forms defined on
. We denote the set of all
-forms as
. For
we usually write
,
,
.
-forms
are just scalar functions
on
.
denotes the constant
-form equal to
everywhere.
Omitted elements of a sequence
When we are given
inputs
and a
-form
we denote omission of the
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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88572a3fac7085591737ebc1ae4a08620cd6423c)
Exterior product
The exterior product is also known as the wedge product. It is denoted by
. The exterior product of a
-form
and an
-form
produce a
-form
. It can be written using the set
of all permutations
of
such that
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)}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/099e17eb01959766ca577c53ac5685c4acab824b)
Directional derivative
The directional derivative of a 0-form
along a section
is a 0-form denoted
Exterior derivative
The exterior derivative
is defined for all
. We generally omit the subscript when it is clear from the context.
For a
-form
we have
as the
-form that gives the directional derivative, i.e., for the section
we have
, the directional derivative of
along
.[6]
For
,[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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f82eabd15dc806524ff92dfd8aab4e55ef30a57)
Lie bracket
The Lie bracket of sections
is defined as the unique section
that satisfies
![{\displaystyle \forall f\in \Omega ^{0}(M)\Rightarrow \partial _{[X,Y]}f=\partial _{X}\partial _{Y}f-\partial _{Y}\partial _{X}f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6efe21ca7603c0a5c45b5cc694bb0dba48bc1502)
Tangent maps
If
is a smooth map, then
defines a tangent map from
to
. It is defined through curves
on
with derivative
such that
![{\displaystyle d\phi (X):=(\phi \circ \gamma )'.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00e460e9248232b1c009108bcaede87ddfa56fe7)
Note that
is a
-form with values in
.
Pull-back
If
is a smooth map, then the pull-back of a
-form
is defined such that for any
-dimensional submanifold
![{\displaystyle \int _{\Sigma }\phi ^{*}\alpha =\int _{\phi (\Sigma )}\alpha .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef2658f9736b6b75491079645602ef7bc769d8f)
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})).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc5b8d660d48c195863b7d1ac2432bb462f9787)
Interior product
Also known as the interior derivative, the interior product given a section
is a map
that effectively substitutes the first input of a
-form with
. If
and
then
![{\displaystyle (\iota _{Y}\alpha )(X_{1},\ldots ,X_{k})=\alpha (Y,X_{1},\ldots ,X_{k}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7432467d40e9b2119890511070ceff9daf4ea97)
Metric tensor
Given a nondegenerate bilinear form
on each
that is continuous on
, the manifold becomes a pseudo-Riemannian manifold. We denote the metric tensor
, defined pointwise by
. We call
the signature of the metric. A Riemannian manifold has
, whereas Minkowski space has
.
Musical isomorphisms
The metric tensor
induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat
and sharp
. A section
corresponds to the unique one-form
such that for all sections
, we have:
![{\displaystyle A^{\flat }(X)=g(A,X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d91920c8abed02798489127a9067f7427b321e)
A one-form
corresponds to the unique vector field
such that for all
, we have:
![{\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdeda4f3918fd92781c95885e92380434a33331)
These mappings extend via multilinearity to mappings from
-vector fields to
-forms and
-forms to
-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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/981fed3d773f82f798871cd40b54b4d3e0b7aa1d)
![{\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 }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a33b6369728242097fd39f7cbb826bf322b66743)
Hodge star
For an n-manifold M, the Hodge star operator
is a duality mapping taking a
-form
to an
-form
.
It can be defined in terms of an oriented frame
for
, orthonormal with respect to the given metric tensor
:
![{\displaystyle ({\star }\alpha )(X_{1},\ldots ,X_{n-k})=\alpha (X_{n-k+1},\ldots ,X_{n}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/433e8594f7a489325fd71ee7f33f0aa10d50c694)
Co-differential operator
The co-differential operator
on an
dimensional manifold
is defined by
![{\displaystyle \delta :=(-1)^{k}{\star }^{-1}d{\star }=(-1)^{nk+n+1}{\star }d{\star }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b07c59d726bca6a8e7961e84b500035a86f46303)
The Hodge–Dirac operator,
, is a Dirac operator studied in Clifford analysis.
Oriented manifold
An
-dimensional orientable manifold M is a manifold that can be equipped with a choice of an n-form
that is continuous and nonzero everywhere on M.
On an orientable manifold
the canonical choice of a volume form given a metric tensor
and an orientation is
for any basis
ordered to match the orientation.
Given a volume form
and a unit normal vector
we can also define an area form
on the boundary
A generalization of the metric tensor, the symmetric bilinear form between two
-forms
, is defined pointwise on
by
![{\displaystyle \langle \alpha ,\beta \rangle |_{p}:={\star }(\alpha \wedge {\star }\beta )|_{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5ced9091ad78010b44b209b9256371c7ffdf1f)
The
-bilinear form for the space of
-forms
is defined by
![{\displaystyle \langle \!\langle \alpha ,\beta \rangle \!\rangle :=\int _{M}\alpha \wedge {\star }\beta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfa8f4593feca617b4ade771ac36e41dbafb6c6)
In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).
Lie derivative
We define the Lie derivative
through Cartan's magic formula for a given section
as
![{\displaystyle {\mathcal {L}}_{X}=d\circ \iota _{X}+\iota _{X}\circ d.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f428b2f5d53f480d19d9c2f9dcc5a140bcd66a0)
It describes the change of a
-form along a flow
associated to the section
.
Laplace–Beltrami operator
The Laplacian
is defined as
.
Important definitions
Definitions on Ωk(M)
is called...
- closed if
![{\displaystyle d\alpha =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d2343c079d5c1a11ea36e2a267e7cc14ae94d43)
- exact if
for some ![{\displaystyle \beta \in \Omega ^{k-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/171e317a4e4f05e6a3bf2cd419d930062e19aa3e)
- coclosed if
![{\displaystyle \delta \alpha =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7748f58052a5c6fd530effb9f72c231976bb2540)
- coexact if
for some ![{\displaystyle \beta \in \Omega ^{k+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b29a4aa543687fad1fc05f997c7b335a059b181)
- harmonic if closed and coclosed
Cohomology
The
-th cohomology of a manifold
and its exterior derivative operators
is given by
![{\displaystyle H^{k}(M):={\frac {{\text{ker}}(d_{k})}{{\text{im}}(d_{k-1})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b221e7d61fcd62f0f09d4401ec52e42d57d8079a)
Two closed
-forms
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c4d4b9f66d69bd60c2cb09b820dff55fc16fec)
A closed surface of genus
will have
generators which are harmonic.
Dirichlet energy
Given
, 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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4efd227d51484f8277c194e10a0707fe8ad78c86)
Properties
Exterior derivative properties
( Stokes' theorem )
( cochain complex )
for
( Leibniz rule )
for
( directional derivative )
for ![{\displaystyle \alpha \in \Omega ^{n}(M),\ {\text{dim}}(M)=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f73e459236c3c4c558607dd337e6f3e51130026f)
Exterior product properties
for
( alternating )
( associativity )
for
( compatibility of scalar multiplication )
( distributivity over addition )
for
when
is odd or
. The rank of a
-form
means the minimum number of monomial terms (exterior products of one-forms) that must be summed to produce
.
Pull-back properties
( commutative with
)
( distributes over
)
( contravariant )
for
( function composition )
Musical isomorphism properties
![{\displaystyle (X^{\flat })^{\sharp }=X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63abbb7c052b0c81fe8bb3e004ac441fffac9a06)
![{\displaystyle (\alpha ^{\sharp })^{\flat }=\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d010b883adc33572a8771e7eadbe081ba11052e0)
Interior product properties
( nilpotent )
![{\displaystyle \iota _{X}\circ \iota _{Y}=-\iota _{Y}\circ \iota _{X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08dcfc4b049a14d3e4250420e9a8c8b4f43f4281)
for
( Leibniz rule )
for ![{\displaystyle \alpha \in \Omega ^{1}(M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094e7092f08e8c7ab50a71036b15ed22e0ce4d87)
for ![{\displaystyle f\in \Omega ^{0}(M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7e6718b820b7bf5622b2707209dc5c5fc3a7c7)
for ![{\displaystyle f\in \Omega ^{0}(M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7e6718b820b7bf5622b2707209dc5c5fc3a7c7)
Hodge star properties
for
( linearity )
for
,
, and
the sign of the metric
( inversion )
for
( commutative with
-forms )
for
( Hodge star preserves
-form norm )
( Hodge dual of constant function 1 is the volume form )
Co-differential operator properties
( nilpotent )
and
( Hodge adjoint to
)
if
(
adjoint to
)
- In general,
![{\displaystyle \int _{M}d\alpha \wedge \star \beta =\int _{\partial M}\alpha \wedge \star \beta +\int _{M}\alpha \wedge \star \delta \beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5cc18898fc32a508893073abd425456f40065de)
for ![{\displaystyle f\in \Omega ^{0}(M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7e6718b820b7bf5622b2707209dc5c5fc3a7c7)
Lie derivative properties
( commutative with
)
( commutative with
)
![{\displaystyle {\mathcal {L}}_{X}(\iota _{Y}\alpha )=\iota _{[X,Y]}\alpha +\iota _{Y}{\mathcal {L}}_{X}\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/988da4012de07faa1652cb0677d01584b026aef1)
( Leibniz rule )
Exterior calculus identities
![{\displaystyle \iota _{X}({\star }\mathbf {1} )={\star }X^{\flat }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1ebe1ca2a30bcc4c25d2d0f911247ad7ab0938)
if ![{\displaystyle \alpha \in \Omega ^{k}(M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb2028497237b33430f1e9e4ebfdfccbd104921e)
![{\displaystyle \iota _{X}(\phi ^{*}\alpha )=\phi ^{*}(\iota _{d\phi (X)}\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/794e37e99d3600188139df1f95273b16b356cedc)
![{\displaystyle \nu ,\mu \in \Omega ^{n}(M),\mu {\text{ non-zero }}\ \Rightarrow \ \exists \ f\in \Omega ^{0}(M):\ \nu =f\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6776adf0dbc82ae83edd29a3d6b47fa238ed75e3)
( bilinear form )
( Jacobi identity )
Dimensions
If
for ![{\displaystyle 0\leq k\leq n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b429c3c44b3dc10332285272eee6f754dbf985)
for ![{\displaystyle k<0,\ k>n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e385de641d0c1f52a2b0a860bca5e74ff55b99e0)
If
is a basis, then a basis of
is
![{\displaystyle \{X_{\sigma (1)}^{\flat }\wedge \ldots \wedge X_{\sigma (k)}^{\flat }\ :\ \sigma \in S(k,n)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13945fdb0754305458db01ca650b219be60b48d4)
Exterior products
Let
and
be vector fields.
![{\displaystyle \alpha (X)=\det {\begin{bmatrix}\alpha (X)\\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab951ef9d0349d96b091f33958202f99adcc229)
![{\displaystyle (\alpha \wedge \beta )(X,Y)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)\\\beta (X)&\beta (Y)\\\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4a3daa6af3a9463c08595e6dbfe745ba4230a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e6f715e2996be4025bd23aa08e2d2cc667c81d)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d004c0821a683d3140eba69714e730840d164627)
Projection and rejection
( interior product
dual to wedge
)
for ![{\displaystyle \alpha \in \Omega ^{k+1}(M),\beta \in \Omega ^{k}(M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c31a51c613947e18a828272739fe03947959ea)
If
, then
is the projection of
onto the orthogonal complement of
.
is the rejection of
, the remainder of the projection. - thus
( projection–rejection decomposition )
Given the boundary
with unit normal vector
extracts the tangential component of the boundary.
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/351c90f62c1aa7bc56098bf5f62caa5d65a03caf)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81200b903c6236e85ab331548fab9d32bc21fa3b)
given a positively oriented orthonormal frame
.
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd448dd63403260a67dc75394c476ab019620187)
Hodge decomposition
If
,
such that[citation needed]
![{\displaystyle \omega =d\alpha +\delta \beta +\gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6e4820331ea22b6f4eca1c736d2f97cec36996)
If a boundaryless manifold
has trivial cohomology
, then any closed
is exact. This is the case if M is contractible.
Relations to vector calculus
Identities in Euclidean 3-space
Let Euclidean metric
.
We use
differential operator
for
.
( scalar triple product )
( cross product )
if ![{\displaystyle \alpha \in \Omega ^{2}(M),\ A=({\star }\alpha )^{\sharp }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9a56eb7816073b754ddef274fa186b41935e7c)
( scalar product )
( gradient )
( directional derivative )
( divergence )
( curl )
where
is the unit normal vector of
and
is the area form on
.
( divergence theorem )
Lie derivatives
(
-forms )
(
-forms )
if
(
-forms on
-manifolds )
if
(
-forms )
![{\displaystyle {\mathcal {L}}_{X}(\mathbf {det} )=({\text{div}}(X))\mathbf {det} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ac1de7169d6dc9afa9f415844c5558cf167345)
References
- ^ Crane, Keenan; de Goes, Fernando; Desbrun, Mathieu; Schröder, Peter (21 July 2013). "Digital geometry processing with discrete exterior calculus". ACM SIGGRAPH 2013 Courses. pp. 1–126. doi:10.1145/2504435.2504442. ISBN 9781450323390. S2CID 168676.
- ^ Schwarz, Günter (1995). Hodge Decomposition – A Method for Solving Boundary Value Problems. Springer. ISBN 978-3-540-49403-4.
- ^ Cartan, Henri (26 May 2006). Differential forms (Dover ed.). Dover Publications. ISBN 978-0486450100.
- ^ Bott, Raoul; Tu, Loring W. (16 May 1995). Differential forms in algebraic topology. Springer. ISBN 978-0387906133.
- ^ Abraham, Ralph; J.E., Marsden; Ratiu, Tudor (6 December 2012). Manifolds, tensor analysis, and applications (2nd ed.). Springer-Verlag. ISBN 978-1-4612-1029-0.
- ^ a b Tu, Loring W. (2011). An introduction to manifolds (2nd ed.). New York: Springer. pp. 34, 233. ISBN 9781441974006. OCLC 682907530.