# Verifying Divergence Theorem

1. Dec 2, 2009

### Darkstalker86

1. The problem statement, all variables and given/known data
Let E be the solid region defined by $$0 \leq z \leq 9+x^2+y^2$$ and $$x^2+y^2 \leq 16$$.
Let S be the boundary surface of E, with positive (outward) orientation.
Also, consider the vector field $$F(x,y,z)=<x,y,x^4+y^4+z>$$

There are five parts to the problem
A) Compute the $$\int\int\int (div F)dV$$

B) Compute the flux of $$\int\int F \bullet dS$$of F, across the (oriented) upper boundary S1 of E.

C) Compute the flux of $$\int\int F \bullet dS$$of F, across the (oriented) side boundary S2 of E.

D) Compute the flux of $$\int\int F \bullet dS$$of F, across the (oriented) bottom boundary S3 of E.

E) Use A,B,C,D to verify the Divergence Theorem for F on E.

2. Relevant equations

3. The attempt at a solution
For A)
The Div F = 3. Also, I converted to cylindrical coords, so the limits of integration would be
$$\\0 \leq z \leq 25; \\0 \leq r \leq 4; \\0 \leq \Theta \leq 2\Pi\\$$

$$\int_{0}^{2\Pi}\int_{0}^{4}\int_{0}^{25}(3r) dz dr d\Theta$$
This integral equals 1200Pi.

For B)
$$\iint_{S1} F \bullet dS$$
I parametrized the boundary of the curve as $h(r, \Theta)=<r\cos\Theta, r\sin\Theta,25>$
Then parametrized the vector field with those parameters.
The partials with respect to each variable of h...
$$h^{}_{r}=<cos\Theta, sin\Theta, 0>$$
and $$h^{}_{\Theta}=<-r\sin\Theta, r\cos\Theta,0>$$
Also, I determined the cross product of $h^{}_{r}X h^{}_{\Theta} = <0,0,r>$

So...(this is after parametrizing and dotting with the normal vector)
Question Here: 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.
$$\int_{0}^{2\Pi}\int_{0}^{4}(25r)*rdrd\Theta$$

I obtained an answer of $$\frac{1600\Pi}{3}$$

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!