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单词 FirstOrderOperatorsInRiemannianGeometry
释义

first order operators in Riemannian geometry


On a pseudo-Riemannian manifoldMathworldPlanetmath M, and in Euclidean space inparticular, one can express the gradient operator, the divergenceoperator, and the curl operator (which makes sense only if M is3-dimensional) in terms of the exterior derivativeMathworldPlanetmath. Let 𝒞(M) denotethe ring of smooth functions on M; let 𝒳(M) denote the𝒞(M)-module of smooth vector fields, and let Ω1(M) denote the𝒞(M)-module of smooth 1-forms. The contractionPlanetmathPlanetmath with the metric tensorg and its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath g-1, respectively, defines the𝒞(M)-module isomorphismsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath

:𝒳(M)Ω1(M),:Ω1(M)𝒳(M).

In local coordinates, this isomorphisms is expressed as

(xi)=jgijdxj,(dxj)=igijxi.

or as the lowering of an index. To wit, for V𝒳(M), we have

V=i=1nVixi,
(V)j=i=1ngijVi,j=1,,n.

The gradient operator, which in tensor notation is expressed as

(gradf)i=gijfxj,f𝒞(M),

can now be defined as

gradf=(df),f𝒞(M).

Another natural structureMathworldPlanetmath on an n-dimensional Riemannian manifold isthe volume formMathworldPlanetmath, ωΩn(M), defined by

ω=detgijdx1dxn.

Multiplication bythe volume form defines a natural isomorphism between functions andn-forms:

ffω,f𝒞(M).

Contraction with thevolume form defines a natural isomorphism between vector fields and(n-1)-forms:

XXω,X𝒳(M),

orequivalently

xi(-1)i+1detgijdx1dxi^dxn,

where dxi^ indicates an omitted factor. The divergenceoperator, which in tensor notation is expressed as

divX=iXi,X𝒳(M)

can be defined in a coordinate-free way by the following relationMathworldPlanetmathPlanetmath:

(divX)ω=d(Xω),X𝒳(M).

Finally, on a 3-dimensional manifold we may define the curloperator in a coordinate-free fashion by means of the following relation:

(curlX)ω=d(X),X𝒳(M).
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更新时间:2025/5/4 10:05:18