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I came across the following question:
Compute the flux out of the hyperboloid $(1) \quad x^2+y^2=1+z^2$ between $z=1$ and $z=-1$ of the force field $\langle x,y,z\rangle$.
The way the question is phrased, I believe one must ignore the disks on the top and bottom of the region.
I proceed in the usual way, using the gradient to find $\text {d} \mathbf S=\frac{\nabla (x^2+y^2-z^2)}{\nabla (x^2+y^2-z^2)\cdot \mathbf k}\text{d}x\text{d}y $ and set up the integral as follows $$\iint_S \langle x, y, z \rangle \cdot \frac {\langle 2x,2y,-2z\rangle}{-2z} \text{d}x\text{d}y=\iint_S -\frac 1z \text{d}x\text{d}y$$ Here we can solve for $z$ in $(1)$ to get that $z=\pm \sqrt {x^2+y^2-1}$. What do we plug into the integral? If I plug in the positive root then we get, after changing to polar coordinates $$\int_0^{2 \pi} \int_1^{\sqrt 2} -\frac {r}{\sqrt{r^2-1}} \text{d}r\text{d}\theta=-2\pi$$ Naturally if I plug in the negative root I get $2\pi$. I get that this discrepancy comes from the fact that I am integrating over the upper and lower halves of the hyperboloid respectively, and that I should add the negative of one of the above results to the other to get the complete answer.
My problem is I do not know which half should have its sign reversed. My biggest problem is I do not know why I don't know.
I thought the above formula for $\text{d} \mathbf S$ was as general as possible, yet I believe that is most likely where the ambiguity is coming from. I think it is reasonable to state that the deepest problem stems from the two branches of the hyperboloid.
So, in general, imagining oneself to be a computer, how would one deal with surfaces with multiple branches (where parametrizations are not always readily available), without using the divergence theorem, and while remaining as systematic as possible?

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