“Exterior calculus identities”的意思、由来-开放百科全书

This article summarizes important 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.

Sections of the tangent bundles, also known as vector fields, are typically denoted as

such that at a point

we have

The boundary of a manifold

is a manifold

, which has dimension

. An orientation on

induces an orientation on

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

denotes the constant

-form equal to

everywhere.

Omitted elements of a sequence

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

such that

Lie bracket

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

.{{citation needed|reason=The notation Xf is only defined in words, and its use here is confusing.|date=November 2018}}

Tangent maps

is a smooth map, then

defines a tangent map from

. It is defined through curves

with derivative

such that

Pull-back

is a smooth map, then the pull-back of a

-form

is defined such that for any

dimensional submanifold

Musical isomorphisms

We call this mapping the sharp operator

and

constitute the musical isomorphisms.

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

The Hodge star operator

is defined as such that it maps

-forms

to their dual

-form

For example, if

is a positively oriented frame for

according to the given metric

, then

We omit to write the dimension

or inversion symbol

with the Hodge star operator as it is evident in the context.

We call

the signature of the metric

. For example in Minkowski space

and in Riemannian manifolds

Co-differential operator

Oriented manifold

-dimensional orientable manifold

is a manifold that can be equipped with a choice of a non-zero

-form

Volume form

On a orientable manifold

the canonical choice of a volume form given a metric

for any positively oriented basis

Area form

Given a volume form

and a unit normal vector

we can also define an area form

on the {{nowrap|boundary

}}

Inner product for k-forms

Lie derivative

We define the Lie derivative

through Cartan's magic formula for a given section

Laplace–Beltrami operator

Important Definitions

Definitions on Ω^k(M)

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.

Dirichlet energy

Properties

Exterior derivative properties

Exterior product properties

Pull-back properties

Musical isomorphism properties

Inner product properties

Hodge star properties

Co-differential operator properties

Lie derivative properties

Exterior calculus identities

Dimensions

Exterior products

Projection and rejection

Sum expressions

Hodge decomposition

Poincaré lemma

has only one cohomology class

and no boundary

, then for any closed

such that

Relations to vector calculus

Identities in Euclidean 3-space

Lie derivatives

References

1. ^{{Cite book |last1=Crane |first1=Keenan |last2=de Goes |first2=Fernando |last3=Desbrun |first3=Mathieu |last4=Schröder |first4=Peter |title=Digital geometry processing with discrete exterior calculus |journal=Proceeding SIGGRAPH '13 ACM SIGGRAPH 2013 Courses |pages=1–126 |date=21 July 2013 |doi=10.1145/2504435.2504442|isbn=9781450323390 }}
2. ^{{cite book |last1=Schwarz |first1=Günter |title=Hodge Decomposition – A Method for Solving Boundary Value Problems |date=1995 |publisher=Springer |isbn=978-3-540-49403-4}}
3. ^{{cite book |last1=Cartan |first1=Henri |title=Differential forms |publisher=Dover Publications |isbn=978-0486450100 |edition=Dover}}
4. ^{{cite book |last1=Bott |first1=Raoul |last2=Tu |first2=Loring W. |title=Differential forms in algebraic topology |publisher=Springer |isbn=978-0387906133}}
5. ^{{cite book |last1=Abraham |first1=Ralph |last2=J.E. |first2=Marsden |last3=Ratiu |first3=Tudor |title=Manifolds, tensor analysis, and applications |publisher=Springer-Verlag |isbn=978-1-4612-1029-0 |edition=2nd}}

Notation

Manifold

k-forms

Omitted elements of a sequence

Exterior product

Lie bracket

Exterior derivative

Tangent maps

Pull-back

Musical isomorphisms

Interior product

Hodge star

Co-differential operator

Oriented manifold

Volume form

Area form

Inner product for k-forms

Lie derivative

Laplace–Beltrami operator

Important Definitions

Definitions on Ωk(M)

Cohomology

Dirichlet energy

Properties

Exterior derivative properties

Exterior product properties

Pull-back properties

Musical isomorphism properties

Inner product properties

Hodge star properties

Co-differential operator properties

Lie derivative properties