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Unformatted text preview: F around the closed path C equals the surface integral of curl F over S . Write âˆ«âˆ« S (curl F ) â‹… d S = âˆ«âˆ« S (curl F ) â‹… n dS = âˆ«âˆ« S âŒ© 1,5,0 âŒª â‹… âŒ© a , b , c âŒª /sqrt( a 2 + b 2 + c 2 ) dS = âˆ«âˆ« S ( a +5 b )/sqrt( a 2 + b 2 + c 2 ) dS , which equals ( a +5 b )/sqrt( a 2 + b 2 + c 2 ) times the area A S ; combining this with our formula A S = ( A D / c ) sqrt( a 2 + b 2 + c 2 ) we see that the original line integral equals ( a +5 b )/ c  times A D . Section 13.9: [Group Work: Group Work 1] ..?. . 1. Since div F > 0, the Divergence Theorem tells us that the net flux is positive. 2. âˆ« â€“1 1 âˆ« â€“1 1 âˆ« â€“sqrt(1â€“ yy ) sqrt(1â€“ yy ) ( x 2 + y 2 ) dx dy dz = âˆ« â€“1 1 âˆ« 2 Ï€ âˆ« 1 ( r 2 ) r dr d Î¸ dz = ( âˆ« â€“1 1 dz ) ( âˆ« 2 d ) ( âˆ« 1 r 3 dr ) = (2)(2 )(1/4) = ....
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 Fall '11
 Staff
 Vector Calculus, Vector field, Stokes' theorem

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