Exterior calculus identities

Exterior calculus identities, or identities in exterior calculus, is mathematical notation used in differential geometry.

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, 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.

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 Ω^''k''(''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