# Verify Divergence Theorem

## Homework Statement

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

## 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 $$\iint_G \textbf{F} \bullet d\textbf{S}$$

## The Attempt at a Solution

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

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

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

However, when I solve this, I get $$\frac{2\pi}{3}$$ when it should be $$\frac{5\pi}{2}$$.

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

vela
Staff Emeritus
I'll note that when I calculated the surface integral and the volume integral, I found them both equal to $4\pi/3$, not $5\pi/2$.