differential geometry w notes from teacher_Part_50

# differential geometry w notes from teacher_Part_50 - 99...

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4.3. STOKES’S THEOREM 99 4.3 Stokes’s Theorem 4.3.1 Orientation of the Boundary Let M be an n -dimensional orientable manifold with boundary M , which is an ( n - 1)-dimensional manifold without boundary. Let M be oriented. Then an orientation on M naturally induces an orientation on M . Let p M and { e 2 , . . . , e n } be a basis in T p M . Let N T p M be a tangent vector at p that is transverse to M and points out of M . Then { N , e 2 , . . . , e n } forms a basis in T p M . Then, by deﬁnition, the basis { e 2 , . . . , e n } has the same orientation as the basis { N , e 2 , . . . , e n } . That is, { e 2 , . . . , e n } is positively oriented in M if { N , e 2 , . . . , e n } is positively oriented in M . 4.3.2 Stokes’ Theorem Theorem 4.3.1 Let M be an n-dimensional manifold and V be a p- dimensional compact oriented submanifold with boundary V in M. Let ω Λ p - 1 M be a smooth ( p - 1) -form in M. Then Z V

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## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_50 - 99...

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