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单词 DifferentialForm
释义

differential form


1 Notation and Preliminaries.

Let M be an n-dimensional differential manifold. Let TM denotethe manifold’s tangent bundle, C(M) the algebra of smoothfunctionsMathworldPlanetmath, and V(M) the Lie algebra of smooth vector fields. Thedirectional derivativeMathworldPlanetmathPlanetmath makes C(M) into a V(M)module. Using local coordinates, the directional derivative operationMathworldPlanetmathcan be expressed as

v(f)=viif,vV(M),fC(M).

2 Definitions.

Differential forms.

Let A be a C(M) module. An -linear mapping α:V(M)A is said to be tensorial if it is aC(M)-homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, in other words, if it satisfies

α(fv)=fα(v)

for allfor all vector fields vV(M) and functions fC(M).More generally, a multilinear map α:V(M)××V(M)A is called tensorial if it satisfies

α(fu,,v)==α(u,,fv)=fα(u,,v)

for all vector fields u,,v and all functions fC(M).

We now define a differentialMathworldPlanetmath 1-form to be a tensorial linear mappingfrom V(M) to C(M). More generally, for k=0,1,2,, wedefine a differential k-form to be a tensorial multilinear,antisymmetric, mapping from V(M)××V(M) (ktimes) to C(M). Using slightly fancier languagePlanetmathPlanetmath, the above amountsto saying that a 1-form is a sectionPlanetmathPlanetmath of the cotangent bundle T*M=Hom(TM,), while a differential k-form as a section ofHom(ΛkTM,).

Henceforth, we let Ωk(M) denote the C(M)-module ofdifferential k-forms. In particular, a differential 0-form is thesame thing as a function. Since the tangent spacesPlanetmathPlanetmath of M aren-dimensional vector spacesMathworldPlanetmath, we also have Ωk(M)=0 for k>n.We let

Ω(M)=k=0nΩk(M)

denote the vector space of all differential forms. There is a naturaloperator, called the exterior product, that endows Ω(M) withthe structureMathworldPlanetmath of a graded algebra. We describe this operation below.

Exterior and Interior Product.

Let vV(M) be a vector field and αΩk(M) adifferential form. We define ιv(ω), the interior productof v and α, to be the differential k-1 form given by

ιv(α)(u1,,uk-1)=α(v,v1,,vk-1),v1,,vk-1V(M).

Theinterior product of a vector field with a 0-form is defined to bezero.

Let αΩk(M) and βΩ(M) be differentialforms. We define the exterior, or wedge productαβΩk+(M) to be theunique differential form such that

ιv(αβ)=ιv(α)β+(-1)kαιv(β)

for all vector fields vV(M). Equivalently, we could have defined

(αβ)(v1,,vk+)=πsgn(π)α(vπ1,,vπk)β(vπk+1,,vπk+),

where the sum is takenover all permutationsMathworldPlanetmath π of {1,2,,k+} such that π1<π2<πk and πk+1<<πk+, and wheresgnπ=±1 according to whether π is an evenor odd permutationMathworldPlanetmath.

Exterior derivative.

The exterior derivative is afirst-order differential operatorMathworldPlanetmath d:Ω*(M)Ω*(M), that can be defined as the unique linear mappingsatisfying

d(dα)=0,αΩk(M);
ιV(df)=v(f),vV(M),fC(M);
d(αβ)=d(α)β+(-1)kαd(β),αΩk(M),βΩ(M).

3 Local coordinates.

Let (x1,,xn) be a system of local coordinates on M, andlet 1,,n denote the corresponding frame ofcoordinate vector fields. In other words,

i(xj)=δi,j

where the right hand side is theusual Kronecker deltaMathworldPlanetmath symbol. By the definition of theexterior derivative,

ιi(dxj)=δi;j

In other words, the 1-forms dx1,,dxn form the dual coframe.

Locally, the i freely generate V(M), meaningthat every vector field vV(M) has the form

v=vii,

where the coordinate componentsMathworldPlanetmath vi are uniquely determined as

vi=v(xi).

Similarly, locally the dxi freely generate Ω1(M). Thismeans thateveryone-form αΩ1(M) takes the form

α=αidxi,

where

αi=ιi(α).

More generally, locally Ωk(M) is a freely generated by thedifferential k-forms

dxi1dxik,1i1<i2<<ikn.

Thus, a differential form αΩk(M) is given by

α=i1<<ikαi1ikdxi1dxik,(1)
=1k!αi1ikdxi1dxik,

where

αi1ik=α(i1,,ik).

Consequently, for vector fields u,v,,wV(M), we have

α(u,v,,w)=αi1i2ikui1vi2wik.

In terms of local coordinates and the skew-symmetrization indexnotation, the interior and exterior product, and the exteriorderivative take the following expressions:

(ιv(α))i1ik=vjαji1ik,vV(M),αΩk+1(M);(2)
(αβ)i1ik+=(k+k)α[i1ikβik+1ik+],αΩk(M),βΩ(M);(3)
(dα)i0i1ik=(k+1)[i0αi1ik],αΩk(M).(4)

Note that some authors prefer a different definition of the componentsof a differential. According to this alternate convention, a factorof k! placed before the summation sign in(1), and the leading factors are removed from(3) and (4).

Titledifferential form
Canonical nameDifferentialForm
Date of creation2013-03-22 12:44:46
Last modified on2013-03-22 12:44:46
Ownerrmilson (146)
Last modified byrmilson (146)
Numerical id28
Authorrmilson (146)
Entry typeDefinition
Classificationmsc 58A10
Definesexterior derivative
Defines1-form
Definesexterior product
Defineswedge product
Definesinterior product
Definestensorial
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