Verify Divergence Theorem

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Homework Statement



Let the surface, G, be the paraboloid [tex]z = x^2 + y^2[/tex] be capped by the disk [tex]x^2 + y^2 \leq 1[/tex] in the plane z = 1. Verify the Divergence Theorem for [tex]\textbf{F}(x,y,z) = 2x\textbf{i} - yz\textbf{j} + z^2\textbf{k}[/tex]

Homework Equations



I have solved the problem using the divergence theorem, that is no problem. However, I am having trouble verifying, where I used the formula [tex]\iint_G \textbf{F} \bullet d\textbf{S}[/tex]

The Attempt at a Solution



My projection on the xy-plane is a circle with the equation [tex]x^2 + y^2 = 1[/tex]. My n1 (normal vector), pointing from the plane z=1, is simply k. The n2, coming out of the surface [tex]z = x^2 + y^2[/tex], I got is 2x i + 2y j - k.

I converted my limits to polar coordinates, to get an integral

[tex]\int_0^{2\pi}\int_0^1 (4r^3(1+cos2\theta) - r^5(1-cos2\theta)) dr d\theta [/tex]

However, when I solve this, I get [tex]\frac{2\pi}{3}[/tex] when it should be [tex]\frac{5\pi}{2}[/tex].

Any ideas on what I have done wrong? thanks for the help
 

Answers and Replies

  • #2
vela
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It's hard to say when you don't show intermediate steps. Your normals look fine.

I'll note that when I calculated the surface integral and the volume integral, I found them both equal to [itex]4\pi/3[/itex], not [itex]5\pi/2[/itex].
 

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