Consider the vector field $\mathbf{F} : \Bbb{R}^3 \to \Bbb{R}^3, \mathbf{F}(x, y, z) = (2y, -z^2, 3x).$
a) Compute the flux of $\mathbf{F}$ through the surface $S$ given by $S = \{{(x, y, z) : x^2 + y^2 + z^3 + z = 1, z \ge 0}\}$ and oriented upwards.
b) Compute the flux of $\nabla \times \mathbf{F}$ through $S$ (defined in part a)).
To answer these, I would use Gauss's Theorem and Stokes' Theorem for parts a) and b), respectively. However, for both of these, I need to determine what the boundary of $S$ is, which I don't know how to do without assistance of graphing software. How could I visualize $S,$ as well as surfaces like $S$ which aren't standard quadric surfaces (spheres, hyperboloids, paraboloids, ellipsoids, etc.) in order to be able to solve problems like these?
Let $f(w)=w+w^3$. This is an increasing function on $\mathbb{R}$, so it has an inverse, and you can obtain the graph $y=f^{-1}(x)$ by reflection of the graph $y=f(x)$ across the line $y=x$. (Note that $f(w) > w$ for $w>0$, so $0 < f^{-1}(w) < w$ for $w>0$.)
This means that you can solve the equation $x^2+y^2+z+z^3=1$ for $z$: $$ z = f^{-1}(1-x^2-y^2) , $$ where $z$ is nonnegative iff $1-x^2-y^2$ is.
Now introduce a radial variable $r=\sqrt{x^2+y^2}$ and draw the curve $z=f^{-1}(1-r^2)$ in the $rz$-plane for $0 \le r \le 1$. (Start with the curve $z=1-r^2$ and move each point $(r,z)$ on that curve, for $0\le r < 1$, a bit downwards, since $0 < f^{-1}(w) < w$ for $w>0$; see plot on Wolfram Alpha).
Then rotate this curve around the $z$ axis to obtain the surface $S$.