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Unformatted text preview: 4.3. STOKESS THEOREM 103 Gauss formula. Let D be a region in a 3dimensional 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 2form in M . Then Z 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 = Z 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 3dimensional 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 1form in M . Then Z 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 !...
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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.
 Spring '10
 Wong
 Geometry

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