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I've got to calculate the flux of the vector field $\mathbf{V} = \nabla \times (-y, x, 0) = (0, 0, 2)$ through a spherical cap $E = \{ (x, y, z): x^2 + y^2 + (z - 4)^2 = 25, z \geq 0. \}$. So I say, using the Divergence Theorem we have: $$Flux \mathbf{V} = \iint_{\partial E } \mathbf{V} \cdot d \mathbf{S} = \iiint_{E} div (\mathbf{V}) \hspace{.1cm} dV = \iiint_{E} 0 \hspace{.1cm} dV = 0.$$

My question is: is what I just did correct?

This problem is supposed to be taken from a final exam in a course a couple semesters ago, but seems a little to easy, that's why I'm hesitant.

UPDATE:

Okay, so the spherical cap has equation $z = 4 + \sqrt{25 - x^2 - y^2}.$ Then $$\iint_{\partial E } \mathbf{V} \cdot d \mathbf{S} = \iint_{D} (- 0 \frac{\partial z}{\partial x} - 0 \frac{\partial z}{\partial y} + 2) dA = \int_{0} ^{2 \pi} \int_{0} ^{5} 2 r dr d\theta = 50\pi.$$ Is this right?

UPDATE 2:

I think something in the statement of the problem might be important. The problem has context: the spherical cap represents a hot air balloon and the hot air escapes through the balloon's porous surface according to the speed field $\mathbf{V}.$

Maybe the word escapes is important. I'm sorry I didn't write it from the beginning.

Chirs Erickson
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  • What exactly is a spherical cap? You seem to have your dimensions all messed up. – Ted Shifrin May 19 '14 at 02:55
  • @TedShifrin The surface defined by $x^2 + y^2 + (z - 4)^2 = 25$ and $z \geq 0.$ – Chirs Erickson May 19 '14 at 03:00
  • If the vector field $ \ \mathbf{V} \ $ is in fact being defined by the curl of another vector field, then we can immediately apply the identity $ \ \nabla \ \cdot \ ( \ \nabla \ \times \ \mathbf{V} \ ) \ = \ 0 \ $ , without even bothering to compute the curl. So the question appears to be about knowing the Divergence Theorem and that useful vector identity. (As for whether the question could be that easy, I've seen a multivariable calculus final where five of the questions had results that came to zero...) – colormegone May 19 '14 at 03:00
  • I'll mention that many exam problems in vector calculus with horrible functions or region geometries turn out to have simple resolutions by applying a particular identity or integral theorem, by recognizing that a field is conservative, etc. (Real-world problems are not always so obliging...) – colormegone May 19 '14 at 03:06
  • Why is your upper limit for the radius 5? – colormegone May 19 '14 at 03:48
  • Because I'm taking $D$ as the projection of the spherical cap. – Chirs Erickson May 19 '14 at 03:51
  • What is the radius of the circle in the $ \ xy-$ plane? That is the "base" of the "cap". (While there is flux entering "upward" through the "overhanging" part of the cap, that is counted as part of the net flux through the "cap" alone. So you still just want the flux through the smaller circle in the $ \ z \ = \ 0 \ $ plane.) – colormegone May 19 '14 at 03:53
  • The radius is $3.$ Should I change the limits in the integral? – Chirs Erickson May 19 '14 at 03:59
  • The context may change matters, depending on what is meant by "escapes". If they meant the hot air escapes only through the upper hemisphere, then you would likely integrate out to a radius of 5. But if the intent is to find the net flux, and air is considered to flow into the enclosed volume through the lower hemispherical surface, as well as the base, then the integration would go out only to radius 3, for the reason I described in my previous comment. – colormegone May 19 '14 at 04:04

1 Answers1

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Note that the spherical cap is not a closed surface, i.e., does not bound a region. In order to apply the Divergence Theorem, you must add in the "base" — the portion of the xy-plane of radius $3$. So the answer is most definitely not $0$.

Ted Shifrin
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  • In that case, the result of zero for the volume is still correct, but the flux through the base should still be calculated in order to find the flux through the "cap". (I had interpreted the condition $ \ z \ \ge \ 0 \ $ to mean that the base is included, but I suspect now that you're correct about what the problem wants. Fortunately, the remaining work is simple.) – colormegone May 19 '14 at 03:10
  • I understand. So I have to calculate the integral $\iint_{\partial E } \mathbf{V} \cdot d \mathbf{S}$ directly. – Chirs Erickson May 19 '14 at 03:17
  • Just on the circular base: the Divergence Theorem shows that the net flux through the "base" plus "cap" is zero. The field at the base is $ \ 2 \ \mathbf{k} \ $ . The flux through the cap will be the negative of the flux you find through the base. – colormegone May 19 '14 at 03:21
  • Be careful with orientations here. What comes in the bottom (flux upward) is what leaves the spherical cap (oriented outward) since there are no sources or sinks inside the region. – Ted Shifrin May 19 '14 at 03:24