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I am trying to solve the following question, which is a standard Surface Integral/Stokes' theorem question. Unfortunately, I've tried to calculate it several times, and each time get the answer slightly wrong. enter image description here

I calculated $\nabla \times B = 3(0,0,x^2+y^2)$

The normal vector for the 'lid' of the shape will be $(0,0,1)$, and for the bottom $(0,0,-1)$

As $\nabla \times B$ only has a non-zero z component, we only need to calculate the z-component of the normal vector on the 'side' of the shape. I calculated this to be $r$, where I used cylindrical co-ordinates to describe the shape in terms of $r$ and $\theta$, as $(r\cos(\theta), r\sin(\theta), r^2)$.

Then using stoke's theorem, I calculated the line integrals to come out as $-\frac{6\pi}{4}(1+\frac{1}{3^4})$, but calculating the surface integral directly I got the top component to come out as $2\pi$, the bottom component to come out as $-6\pi\frac{1}{3^4}$ and the middle component to come out as $\frac{6\pi}{4}(1-\frac{1}{3^4})$

That means I found the surface integral to come out as $\frac{4}{9}\pi$ and the line integral to come out as $\frac{41}{27}\pi$ which is not the same :(

In any case, I have spent a very long time re-doing the calculations and have made no progress, so thank you very much

riemann_lebesgue
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  • It's unclear from what you have written if you did this, but the surface does not include the "lids". It is only the paraboloid band. There are still two line integrals, however. And as a sanity check, the normal vector for the paraboloid should be parallel to $\langle 2x, 2y, -1 \rangle$ – Ninad Munshi Aug 28 '20 at 16:01
  • @NinadMunshi I thought Stokes' Theorem only applied to closed surfaces? – riemann_lebesgue Aug 28 '20 at 16:21
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    You have it confused with divergence theorem in that regard. If a surface was closed, how could it ever have a line boundary? – Ninad Munshi Aug 28 '20 at 16:46
  • oh! thank you so much. I am self teaching the material and whenever I've calculated a surface integral it's gone wrong! – riemann_lebesgue Aug 29 '20 at 17:25
  • presumably if the surface is closed, then we can use Stoke's theorem to show it (curl(B)) evaluates to 0 – riemann_lebesgue Aug 29 '20 at 17:26
  • in any case, thank you very much!! That makes a lot more sense. I suppose I thought the boundary was what intuitively looked like the boundary - where the surface changed from the paraboloid to the flat sheet in this case – riemann_lebesgue Aug 29 '20 at 17:27

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