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 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 k-forms

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

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

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]

Lie bracket

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

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

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

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:

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

These mappings extend via multilinearity to mappings from -vector fields to -forms and -forms to -vector fields through

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 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.

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 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 , its Dirichlet energy is

Properties

Exterior derivative properties

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

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

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 )
In general,
for

Lie derivative properties

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

Exterior calculus identities

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 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
( 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 )

References

  1. 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.
  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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