“Stokes' Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Stokes' Theorem

释义

Stokes' Theorem

For a Differential k-Form with compact support on an oriented -dimensional Manifold ,

(1)

where

is the Exterior Derivative of the differential form

. This connects to the ``standard''Gradient, Curl, and Divergence Theoremsby the following relations. If

is a function on

(2)

where

(the dual space) is the duality isomorphism between a Vector Spaceand its dual, given by the Euclidean Inner Product on

. If

is a Vector Field on a

(3)

where

is the Hodge Star operator. If

is a Vector Field on

(4)

With these three identities in mind, the above Stokes' theorem in the three instances is transformed into theGradient, Curl, and Divergence Theoremsrespectively as follows. If is a function on and is a curve in , then

(5)

which is the Gradient Theorem. If

is a Vector Field and

an embedded compact3-manifold with boundary in

, then

(6)

which is the Divergence Theorem. If

is a Vector Field and

is an oriented, embedded, compact 2-Manifold with boundary in

, then

(7)

which is the Curl Theorem.

Physicists generally refer to the Curl Theorem

(8)

as Stokes' theorem.

See also Curl Theorem, Divergence Theorem, Gradient Theorem

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