Let $E$ be the solid region between the plane $z=4$ and the paraboloid $z=x^2+y^2$. Let $\vec{F} = (-\frac{1}{3}x^3+e^{z^2})\vec{i}+(-\frac{1}{3}y^3+x\tan z)\vec{j}+4z\vec{k}$
a) Compute the flux of $\vec{F}$ outward through the boundary of $E$.
b) Let S be the part of the paraboloid $z=x^2+y^2$ lying below $z=4$ plane oriented so that $\vec{N}$ has a positive $\vec{k}$ component Compute the flux of $\vec{F}$ through $S$.
I don't understand the conceptual idea between these two questions, how are they not asking for the same thing? Also a little confused about what inward flux is and what outward flux is, does this have something to do with the normal vector?
The solid $E$ looks like a "spinning top". Its boundary $\partial E$ consists of a part $S$ of the paraboloid and a circular disc $D$ in the plane $z=4$, both of these (conventionally) oriented outwards. The latter means that the positive normal ${\bf n}$ on $\partial E$ looks towards the outside of $E$; in particular ${\bf n}=(0,0,1)=\vec k$ on $D$.
For problem a) you have to compute two integrals, namely $$\int_{\partial E}\ldots=\int_S\ldots \ +\ \int_D\ldots\ ,$$ whereby you have to make sure that you choose a parametrization of $S$ which creates the right orientation of ${\bf n}$. By the way: You can also use Gauss' divergence theorem, which says that $$\int_{\partial E}\vec F\cdot{\bf n}\>{\rm d}\omega=\int_E{\rm div}(\vec F)\>{\rm d}V\ .$$ Maybe the latter integral is easier to compute than the two surface integrals.
In problem b) they ask for the value of the integral $\ -\int_S\vec F\cdot{\bf n}\>{\rm d}\omega$, where ${\bf n}$ is as before, and I have introduced a minus sign since the posers of the problem (when they say "oriented so that ${\bf n}$ has a positive $\vec k$ component") count the flux upwards, or to the interior of the paraboloid, as positive. For this problem you cannot use Gauss' theorem since $S$ is not the full boundary of some body.