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Verifying Divergence Theorem
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[QUOTE="Darkstalker86, post: 2470771, member: 216940"] [h2]Homework Statement [/h2] Let E be the solid region defined by [tex]0 \leq z \leq 9+x^2+y^2[/tex] and [tex] x^2+y^2 \leq 16[/tex]. Let S be the boundary surface of E, with positive (outward) orientation. Also, consider the vector field [tex]F(x,y,z)=<x,y,x^4+y^4+z>[/tex] There are five parts to the problem A) Compute the [tex]\int\int\int (div F)dV[/tex] B) Compute the flux of [tex]\int\int F \bullet dS[/tex]of F, across the (oriented) upper boundary S[SUB]1[/SUB] of E. C) Compute the flux of [tex]\int\int F \bullet dS[/tex]of F, across the (oriented) side boundary S[SUB]2[/SUB] of E. D) Compute the flux of [tex]\int\int F \bullet dS[/tex]of F, across the (oriented) bottom boundary S[SUB]3[/SUB] of E. E) Use A,B,C,D to verify the Divergence Theorem for F on E. [h2]Homework Equations[/h2] [h2]The Attempt at a Solution[/h2] For A) The Div F = 3. Also, I converted to cylindrical coords, so the limits of integration would be [tex]\\0 \leq z \leq 25; \\0 \leq r \leq 4; \\0 \leq \Theta \leq 2\Pi\\[/tex] [tex]\int_{0}^{2\Pi}\int_{0}^{4}\int_{0}^{25}(3r) dz dr d\Theta[/tex] This integral equals 1200Pi. For B) [tex]\iint_{S1} F \bullet dS[/tex] I parametrized the boundary of the curve as [itex]h(r, \Theta)=<r\cos\Theta, r\sin\Theta,25>[/itex] Then parametrized the vector field with those parameters. The partials with respect to each variable of h... [tex]h^{}_{r}=<cos\Theta, sin\Theta, 0>[/tex] and [tex]h^{}_{\Theta}=<-r\sin\Theta, r\cos\Theta,0>[/tex] Also, I determined the cross product of [itex]h^{}_{r}X h^{}_{\Theta} = <0,0,r>[/itex] So...(this is after parametrizing and dotting with the normal vector) [B]Question Here:[/B] There are factors in this integral that I noticed are odd functions over a symmetric surface. That means that when I integrate them, won't they just go to Zero, and I can save myself time and not do them now correct? Assuming that is correct, this is the integral I came up with. [tex]\int_{0}^{2\Pi}\int_{0}^{4}(25r)*rdrd\Theta[/tex] I obtained an answer of [tex]\frac{1600\Pi}{3}[/tex] Now, I'm not exactly sure that is correct. I know I need to follow the same basic steps in order to determine the side flux and bottom flux integrals. I am having trouble figuring out the flux integral for the side boundary. Any assistance would be great! [/QUOTE]
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