Is is incorrect to integrate multiple surfaces when using Stokes Theorem

Question

So far, I have understood that when using Stokes Theorem to find the flux of the curl on a solid that has a boundary, one can use any of the surfaces of that solid.

For example in my book for this problem (Transcendental Functions, Smith and Minton 14.8 #13)

13. $C$ is the boundary of the portion of the paraboloid $y = 4 − x^2 − z^2$ with $y > 0$, $\mathbf n$ to the right, $F = \langle x^2z, 3 \cos y, 4z^3 \rangle$.

the solution is to just integrate over the disc $x^2+z^2 \le 4$.

However, elsewhere it says to integrate multiple surfaces!

For example, here (Transcendental Functions, Smith and Minton 14.8 #23)

$ S$ is the boundary of the solid bounded by the hyperboloid $x^2 + y^2 − z^2 = 4$, $z = 0$ and $z = 2$ with $z < 2$, $\mathbf n$ downward at bottom, $F = \langle 2^y − x \cos x, y^2 + 1, e^{−z^2} \rangle$

it says to integrate two surfaces, the portion of the hyperboloid and the circle with radius 2.

Is there something different about the second problem causing them to integrate over two surfaces, or is there a mistake?

Answer 1

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:

1. The first piece $S_1$ consists of the portion of the hyperboloid $x^2 + y^2 - z^2 = 4$ for $0 \leq z \leq 2$. This piece has as boundary two circles - the circle on the plane $z = 0$ and the circle on the plane $z = 2$.
2. The second piece $S_2$ consists of the disc $x^2 + y^2 \leq 4$ on the plane $z = 0$.

The surface $S$ is the union of $S_1$ and $S_2$ and has as boundary only the circle on the plane $z = 2$. In order to compute the flux using directly from the definition, you need to compute $\operatorname{curl} F$ and then integrate it over $S_1$ and $S_2$. Alternatively, you can use Stokes' theorem and integrate $F$ over the boundary which consists of the circle on the plane $z = 2$.

My guess is based on the fact that you are told that n points downward at the bottom which suggests that the "bottom" (the disc on the plane $z = 0$) is part of the surface while you are also told that $z < 2$ which suggests that the "top" (the disc on the plane $z = 2$) is not part of the surface.

Answer 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 of all of the curves.