electromagnetism - A closed surface, no charge enclosed, yet flux not 0?

http://img213.imageshack.us/img213/5474/gausslaw.jpg!

The book says it is $E_0\pi r^2$ because the flux through the circle is equal to the curved part of the paraboloid.

I don't understand this, shouldn't the total flux be 0 for the whole surface? IN fact, since the E field is constant, then the flux must also be 0.

Answer

Be careful here.

Gauss's law tells you that the flux through the (whole) closed surface is proportional to the enclosed charge and therefore zero in this case.

That's one fact.

The second fact is that you have a constant electric field in this region of space, and that means that the flux through the circular end-cap (which is not a closed surface) is $E_0\pi r^2$.

Now we put the two facts together, the combination of the end-cap plus the parabaloid is a closed surface, which means that (because the flux through the end cap is pointed in and is therefore negative) we get

$$0 = \mathcal{F}_\text{end-cap} + \mathcal{F}_\text{parabaloid} = - E_0 \pi r^2 + \mathcal{F}_\text{parabaloid}$$

or, if we re-arrange things

$$\mathcal{F}_\text{parabaloid} = E_0 \pi r^2 .$$