Divergence Theorem

Divergence Theorem - Divergence Theorem Examples Gauss'...

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Divergence Theorem Examples Gauss' divergence theorem relates triple integrals and surface integrals. GAUSS' DIVERGENCE THEOREM Let be a vector field. Let be a closed surface, and let be the region inside of . Then: F W W e (( ((( a b W F A F † . œ .Z e div EXAMPLE 1 Evaluate , where is the sphere . (( a b W # # # $B  #C † . W B  C  D œ * i j A SOLUTION We could parameterize the surface and evaluate the surface integral, but it is much faster to use the divergence theorem. Since: div a b a b a b a b $B  #C œ $B  #C  ` ` ` `B `C `D i j the divergence theorem gives: (( ((( a b a b W $B  #C † . œ œ ")! i j A e the volume of the sphere 1 è EXAMPLE 2 Evaluate , where is the boundary of the cube defined by (( ˆ W # $ C D  C  BD † . W i j k A " Ÿ B Ÿ " " Ÿ C Ÿ " ! Ÿ D Ÿ # , , and . SOLUTION Since: div ˆ ˆ ˆ ‰ a b C D  C  BD # $ i j k œ C D  C BD œ $C  B ` ` ` `B `C `D # $ # the divergence theorem gives: (( ((( ˆ ˆ ( ( ( ˆ ( W # $ # ! " " # " " # " " # C D  C  BD † . œ $C  B .Z œ $C  B .B .C .D œ # 'C .C œ ) i j k A e è
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EXAMPLE 3 Let be the region in bounded by the paraboloid and the plane , e $ # # D œ B  C D œ " and let
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This note was uploaded on 03/01/2012 for the course EE 101 taught by Professor Munk during the Spring '12 term at Alaska Anch.

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Divergence Theorem - Divergence Theorem Examples Gauss'...

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