Exterior calculus identities

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

, 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 bundle

is the tangent bundle of the smooth manifold .

, denote the tangent spaces of , at the points , , respectively.

Sections of the tangent bundles, also known as vector fields, are typically denoted as such that at a point we have .

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 .

k-forms

-forms 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

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

Lie bracket

The Lie bracket of sections is defined as the unique section that satisfies

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 directional derivative -form. i.e. in the direction we have .[6]

For ,[6]

Tangent maps

If is a smooth map, then defines a tangent map from to . It is defined through curves on with derivative such that

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

The pull-back can also be expressed as

Musical isomorphisms

The metric tensor induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat and sharp . A vector field corresponds to the unique one-form such that for all tangent vectors , we have:

This extends via multilinearity to a mapping from -vector fields to -forms through

A one-form corresponds to the unique vector field such that for all , we have:

This mapping similarly extends to a mapping from -forms to -vector fields through

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

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 :

Co-differential operator

The co-differential operator on an dimensional manifold is defined by

The Hodge–Dirac operator, , is a Dirac operator operator studied in Clifford analysis.

Oriented manifold

An -dimensional orientable manifold is a manifold that can be equipped with a choice of an -form that is continuous and nonzero everywhere on .

Volume form

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.

Area form

Given a volume form and a unit normal vector we can also define an area form on the boundary

Bilinear form on k-forms

A generalization of the metric tensor, the symmetric bilinear form between two -forms , is defined pointwise on by

The -bilinear form for the space of -forms is defined by

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

It describes the change of a -form along a flow map associated to the section .

Laplace–Beltrami operator

The Laplacian is defined as .

Important Definitions

Definitions on Ωk(M)

is called...

  • closed if
  • exact if for some
  • coclosed if
  • coexact if for some
  • harmonic if closed and coclosed

Cohomology

The -th cohomology of a manifold and its exterior derivative operators is given by

Two closed -forms are in the same cohomology class if their difference is an exact form i.e.

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

Dirichlet energy

Given

Properties

Exterior derivative properties

( Stokes' theorem )
( cochain complex )
for ( Leibniz rule )
for ( directional derivative )
for

Exterior product properties

for ( alternating )
( associativity )
for ( distributivity 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

Interior product properties

( nilpotent )
for ( Leibniz rule )
for
for
for

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 )
for

Lie derivative properties

( commutative with )
( commutative with )
( Leibniz rule )

Exterior calculus identities

if
if
( bilinear form )
( Jacobi identity )

Dimensions

If

for
for

If is a basis, then a basis of is

Exterior products

Let and be vector fields.

Projection and rejection

( interior product dual to wedge )
for

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

given a positively oriented orthonormal frame .

Hodge decomposition

If , such that

Poincaré lemma

If a boundaryless manifold has trivial cohomology , then for any closed , there exists such that . 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 )
if
( dot product )
( gradient -form )
( 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 )

References

  1. Crane, Keenan; de Goes, Fernando; Desbrun, Mathieu; Schröder, Peter (21 July 2013). Digital geometry processing with discrete exterior calculus. Proceeding SIGGRAPH '13 ACM SIGGRAPH 2013 Courses. pp. 1–126. doi:10.1145/2504435.2504442. ISBN 9781450323390.
  2. Schwarz, Günter (1995). Hodge Decomposition – A Method for Solving Boundary Value Problems. Springer. ISBN 978-3-540-49403-4.
  3. Cartan, Henri (26 May 2006). Differential forms (Dover ed.). Dover Publications. ISBN 978-0486450100.
  4. Bott, Raoul; Tu, Loring W. (16 May 1995). Differential forms in algebraic topology. Springer. ISBN 978-0387906133.
  5. 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.
  6. Tu, Loring W. (2011). An introduction to manifolds (2nd ed.). New York: Springer. pp. 34, 233. ISBN 9781441974006. OCLC 682907530.
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