2
Stokes's theorem in its simplest form applies to a single closed curve, and (any) surface interpolating the curve. But it's trivial to extend to the case where you have multiple closed curves (as in the second problem): just apply Stokes's theorem separately to each component.
Or, if you want to get really fancy, apply Stokes's theorem to a Siefert surface ...
2
I suspect that the source of your confusion is the somewhat awkward description of the surface $S$ in the second problem. My guess is that you are asked to compute the flux of $\operatorname{curl} F$ on the surface $S = S_1 \cup S_2$ which consists of two pieces:
The first piece $S_1$ consists of the portion of the hyperboloid $x^2 + y^2 - z^2 = 4$ for $0 \...
2
Suppose $f$ is already linear: $f(x) = Ax$. You would like $f'(x) = A$.
what about we require that (4) goes to zero faster than some other function? How about faster than $\sqrt{|h|}$...
Then any number of functions $f'(x)$ satisfy your definition, e.g. $f'(x) = 0$.
or $|h|^2$ or $|h|^3$?
Then your $f'(x)$ does not exist.
Additionally I'm not ...
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