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differential geometry w notes from teacher_Part_52

differential geometry w notes from teacher_Part_52 - 103...

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4.3. STOKES’S THEOREM 103 Gauss formula. Let D be a region in a 3-dimensional manifold M and U be the parameter preimage of D in R 3 . Let D be the boundary of D and U be the boundary of U . Let F be a 2-form in M . Then U F 12 x 3 + F 23 x 1 + F 31 x 2 ( x 1 , x 2 , x 3 ) ( u 1 , u 2 , u 3 ) du 1 du 2 du 3 = U F 12 ( x 1 , x 2 ) ( z 1 , z 2 ) + F 23 ( x 2 , x 3 ) ( z 1 , z 2 ) + F 31 ( x 3 , x 1 ) ( z 1 , z 2 ) dz 1 dz 2 Stokes’ formula. Let S be a surface in a 3-dimensional manifold M and S be its boundary. Let U be the preimage of S in the parameter plane R 2 and U = [ a , b ] be its boundary. Let A be a 1-form in M . Then U A 2 x 1 - A 1 x 2 ( x 1 , x 2 ) ( u 1 , u 2 ) + A 3 x 1 - A 1 x 3 ( x 1 , x 3 ) ( u 1 , u 2 ) + A 3 x 2 - A 2 x 3 ( x 2 , x 3 ) ( u 1 , u 2 ) du 1 du 2 = b a A 1 dx 1 dt + A 2 dx 2 dt + A 3 dx 3 dt dt di ff geom.tex; April 12, 2006; 17:59; p. 104
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104 CHAPTER 4. INTEGRATION OF DIFFERENTIAL FORMS 4.4 Poincar´e Lemma Definition 4.4.1 Let M be a manifold.
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