Can You Solve the Divergence Theorem for a Cylindrical Vector Field?

[Baumer8993]

Homework Statement

Verify the divergence theorem by computing both integrals for the vector field
F = <x^3, y^3, z^2> over a cylindrical region define by x^2+y^2 ≤ 9.

Homework Equations

Divergence Theorem, and Flux Integrals.

The Attempt at a Solution

I did the divergence theorem, and got 279 pi for my answer. I did the integral in cylindrical, and regular with the same answer so that I know is correct. I know I need to break this up into three different surfaces. I can do the top, and the bottom easily since N = <0, 0, 1> for the top, and N = < 0, 0, -1> for the bottom, then I just plug in the z value for it.

The side of the cylinder is giving me trouble. I have looked up ways to do it online, but they make no sense, or are disorganized. Where do I start, and then where do I go from there?

 

[HallsofIvy]

You haven't completely stated the problem. You say it is over the cylidrical region [itex]x^2+ y^2\le 9[/itex] but don't say what the ends are. What values of z?

You want to integrate over the surface of the cyinder. Any point on the circle [itex]x^2+ y^2= 9[/itex] can be written [itex](3cos(\theta), 3sin(\theta))[/tiex] with [itex]\theta[/itex] going from 0 to [itex]2\pi[/itex]. They cylinder has axis along the z-axis so we can take any point on the cylinder as given by [itex]\vec{r}(\theta, z)= 3cos(\theta)\vec{i}+ 3sin(\theta)\vec{j}+ z\vec{k}[/itex].

The [itex]\vec{r}_\theta= -3sin(\theta)\vec{i}+ 3cos(\theta)\vec{j}[itex] and [itex]\vec{r}_z= \vec{k}[/itex] and the cross product of those vectors gives the vector differential of surface area, [itex](3cos(\theta)\vec{i}+ 3 sin(\theta)\vec{j})d\theta dz[/itex].

The two ends will have vector differential [itex]\vec{k} drd\theta[/itex] and [itex]-\vec{k}drd\theta[/itex] with r going from 0 to 3 and [itex]\theta[/itex] going from 0 to [itex]2\pi[/itex]