This article summarizes several identities in exterior calculus, a mathematical calculus used in differential geometry.
Notation The following notation is 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 .
For ,
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 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
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