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5 Pages

### GaussExamples

Course: MATH 601, Fall 2009
School: Bard College
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Word Count: 924

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Theorem Divergence Examples Gauss' divergence theorem relates triple integrals and surface integrals. GAUSS' DIVERGENCE THEOREM Let F be a vector field. Let W be a closed surface, and let e be the region inside of W . Then: ( ( F . A oe ( ( ( divaFb .Z W e EXAMPLE 1 Evaluate ( ( a\$B i #C jb . A, where W is the sphere B# C# D # oe *. W SOLUTION We could parameterize the surface and evaluate the surface...

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Theorem Divergence Examples Gauss' divergence theorem relates triple integrals and surface integrals. GAUSS' DIVERGENCE THEOREM Let F be a vector field. Let W be a closed surface, and let e be the region inside of W . Then: ( ( F . A oe ( ( ( divaFb .Z W e EXAMPLE 1 Evaluate ( ( a\$B i #C jb . A, where W is the sphere B# C# D # oe *. W SOLUTION We could parameterize the surface and evaluate the surface integral, but it is much faster to use the divergence theorem. Since: diva\$B i #C jb oe ` ` ` a\$Bb a#C b a! b oe & `B `C `D the divergence theorem gives: ( ( a\$B i #C jb . A oe ( ( ( & .Z oe & , athe volume of the sphereb oe ")!1 W e EXAMPLE 2 Evaluate ( ( ^C # D i C \$ j BD k . A, where W is the boundary of the cube defined by W " Y B Y ", " Y C Y ", and ! Y D Y #. SOLUTION Since: div^C # D i C \$ j BD k oe ` # ` \$ ` ^C D ^C aBD b oe \$C# B `B `C `D the divergence theorem gives: # \$ # ( ( ^C D i C j BD k . A oe ( ( ( ^\$C B .Z W e oe ( ( ( ^\$C # B .B .C .D # " " ! " " oe #( 'C # .C oe ) " " EXAMPLE 3 Let e be the region in `\$ bounded by the paraboloid D oe B# C # and the plane D oe ", and let W be the boundary of the region e. Evaluate ( ( ^C i Bj D # k . A. W SOLUTION Here is a sketch of the region in question: z z=1 (1, 1) z=r r 2 Since: div^C i Bj D # k oe the divergence theorem gives: # ( ( D k . A oe ( ( ( #D .Z W e ` ` ` # ^D oe #D aC b aB b `B `C `D It is easiest to set up the triple integral in cylindrical coordinates: ( ( ( #D .Z oe ( e #1 ! ( ( #D < .D .< . ) ! <# " " Doe<# " " oe #1 ( 'D # <" ! " .< oe #1( ^< <& .< ! " " #1 oe #1 OE oe # ' \$ In general, you should probably use the divergence theorem whenever you wish to evaluate a vector surface integral over a closed surface. The divergence theorem can also be used to evaluate triple integrals by turning them into surface integrals. This depends on finding a vector field whose divergence is equal to the given function. EXAMPLE 4 Find a vector field F whose divergence is the given function 0 aBb. (a) 0 aBb oe " (b) 0 aBb oe B# C (c) 0 aBb oe B# D # SOLUTION The formula for the divergence is: divaFb oe f F oe `JC `JB `JD `B `C `D We get to choose JB , JC , and JD , so there are several possible vector fields with a given divergence. (This is similar to the freedom enjoyed when finding a vector field with a given rotation.) (a) F oe Bi works, as does F oe C j, F oe D k, and so forth. (b) Three possible solutions are F oe (c) It is difficult to integrate B# D # with respect to B or , D but we can integrate with respect to C to get F oe C B# D # j. EXAMPLE 5 Let e be the region defined by B# C # D # Y ". Use the divergence theorem to e " \$ B C i, \$ Foe " # # B C j, # and F oe B# CD k. evaluate ( ( ( D # .Z . SOLUTION Let W be the unit sphere B# C # D # oe ". By the divergence theorem: # ( ( ( D .Z oe ( ( F . A e W where F is any vector field whose divergence is D # . One possible choice is F oe # ( ( ( D .Z oe ( ( e " \$ D k: \$ W " \$ D k .A \$ All that remains is to compute the surface integral ( ( z " \$ D k . A. W \$ We have parameterized the sphere many times by now: >oe) r B oe cos ? cos > so C oe cos ? sin > D oe sin ? and 1 1 Y?Y # # < oe cos ? D oe sin ? ! Y > Y #1 This gives: i `B . A oe `> `B `? j `C `> `C `? k i j k `D ! .> .? .> .? oe cos ? sin > cos ? cos > `> sin ? cos > sin ? sin > cos ? `D `? oe ^ cos# ? cos > cos# ? sin > cos ? sin ? .> .? so: " \$ " #1 1# asin ?b\$ cos ? sin ? .? .> D k .A oe ( ( (( \$ \$ ! 1# W oe #1 1# sin% ? cos ? .? ( \$ 1# #1 " & " sin ? \$ & 1# %1 "& 1# oe oe Of course, in the last example it would have been faster to simply compute the triple integral. In reality, the divergence theorem is only used to compute triple integrals that would otherwise be difficult to set up: EXAMPLE 6 Let W be the surface obtained by rotating the curve < oe cos ? D oe sin #? around the D -axis: z 1 1 Y?Y # # r Use the divergence theorem to find the volume of the regi...

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Wisconsin - ENGR - 724
Wisconsin - ENGR - 724
Wisconsin - ENGR - 724
Wisconsin - ENGR - 724
Wisconsin - ENGR - 724
Wisconsin - ENGR - 724
Wisconsin - ENGR - 724
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