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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  ! œ & ` ` ` `B `C `D i j the divergence theorem gives: (( ((( a b a b W \$B #C †. œ &.Z 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 "ŸBŸ" "ŸC Ÿ" !ŸD 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 œB C D œ" and let be the boundary of the region . Evaluate . W C B D †. e (( ˆ W
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