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Hello and how you doing today? I just came across a problem which need to use Stokes theorem.

The problems says:

Evaluate the surface integral

$$ \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} $$

where F(x,y,z)=$(y^2)i$ + $(2xy)j$+$(xz^2)k$ and S is the surface of the paraboloid $z= x^2+y^2$ bounded by the planes $x=0,y=0$ and $z=1$ in the first quadrant pointing upward.

I got $\nabla F$ which is $(-z^2)j$

So, I stuck at here because I dont know the boundary C in order to use Stokes theorem. So could someone please help me to start?

By the way thank you very much for taking your time and consideration to help me on this problem.

enter image description here

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Plotting surfaces, it's pretty much clear that we will have the following figure. enter image description here

Stokes theorem states that the path integral of a vector field around those dotted line is equal to the the surface vector integral of the curl of the field bound by surface. i.e. $$ \oint_\gamma \vec F \cdot dr = \iint_\Omega \nabla \times \vec F \cdot \hat n d\sigma $$

First, let's calculate the left part, the path integral around along the curve going by route $OA \to AB \to BO$.

Along $OA$, the curve is $z=x^2$ and $y=0$. $$\int_{OA} ( y^2 \hat i + 2xy \hat j + xz^2 \hat k) \cdot dr = \int_{OA} xz^2 \hat k \cdot (dx \hat i + dz \hat k ) = \int_{OA} xz^2 dz $$ Putting $z=x^2$ and $x = 0 \to 1$, we get $$ \int_0^1 2x^6dx = \frac 2 7 \hspace{1cm} (1)$$

Now along AB, $z=1$, constant. $$\int_{AB} ( y^2 \hat i + 2xy \hat j + xz^2 \hat k) \cdot dr = \int_{AB} ( y^2 \hat i + 2xy \hat j + xz^2 \hat k) \cdot ( dx \hat i + dy \hat j)$$

The circular curve can be parametrize $x = \cos(\theta)$ and $y = \sin(\theta)$, and we know that in along the curve $\theta = 0 \to \frac \pi 2 $, and that gives $dx = -\sin\theta d\theta$ and $dy = \cos\theta d\theta$, and simplifying we get

$$\int_0^{\frac \pi 2 } \left( -\sin^3 \theta + 2 \sin\theta \cos^2 \theta \right) d\theta = 0 \hspace{1cm} (2)$$

Along $OB$ we proceed in the same as along $OA$ but here $x=0$ and we get $$\int_{BO} \vec F \cdot dr = 0 \hspace{1cm} (3)$$

Adding $(1), (2)$ and $(3)$, we get $ \displaystyle \oint_\gamma \vec F \cdot dr = \frac 2 7 $.

The right part:: The curl of the field turns out to be $- z^2 \hat j $ and the integral is $$ \iint_\Sigma (-z^2 \hat j)\cdot \hat n ds $$

The surface $\displaystyle z = x^2 + y^2$ can be parameterized as $S = (r \cos\theta, r\sin\theta, r^2)$ where $r =0\to 1$ and $\theta = 0 \to \frac \pi 2$ and the surface integral can be evaluated as $$\int_0^1 \int_0^{\frac \pi 2 } -r^4 \hat j \cdot \left( \frac{\partial S}{\partial r } \times \frac{\partial S}{\partial \theta } \right ) dr \; d\theta $$ The cross product can be calculated as $ -2 r^2 \cos (\theta ) \hat i -2 r^2 \sin (\theta ) \hat j + r \sin ^2(\theta )+r \cos ^2(\theta ) \hat k$

And the integral turns out as

$$ \int_0^1\int_0^{\frac \pi 2 } 2r^6 \sin ( \theta) d\theta dr = \frac 2 7 $$

ADDED:: For plotting surfaces, just plot all surfaces and take the surface of the parabloid that is bounded by $x=0, y=0, z=1$ on first quadrant, that is the one on the first quadrant which looks like the strip above. enter image description here

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  • $\begingroup$ Thank you very much experimentX for your great effort and clear explanation. could I know which software are you using to plot the graph because I would like to use it for other problems also.And because of your help, I give +50 reputation.cheers $\endgroup$ – Garett Sep 24 '13 at 5:18
  • $\begingroup$ I forget to tell you. Isnt the top surface should be closed surface since it is bounded by z=1.Since in the diagram it is hollow on top. And I'm not sure I'm correct. Correct me if I'm wrong. $\endgroup$ – Garett Sep 24 '13 at 5:54
  • $\begingroup$ @Garett $z=y^2+x^2$ is an infinite parabloid. $z=1$ is a plane that cuts of that infinite parabloid. We are only asked to integrate along that surface of parabloid that is bound by $x=0, y=0$ ,,, and $z=1$, these planes cut the infinite parabloid into that strip in picture. $\endgroup$ – Santosh Linkha Sep 24 '13 at 5:58
  • $\begingroup$ @Garett also I used Mathematica for Plotting. if you can parametrize any surface then you can plot it very easily. Here is one way of plotting it. $\endgroup$ – Santosh Linkha Sep 24 '13 at 5:59
  • $\begingroup$ Thank you for your kind help to correct me out my mistake and tips for the plotting $\endgroup$ – Garett Sep 24 '13 at 6:02
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It's readily to see it when you draw a picture of $z=x^2+y^2$ intersected with three planes. The boundary contains three parts:

  • $(0,0,0)\rightarrow(0,1,1)$ along $z=y^2$ for $x=0$

  • $(0,1,1)\rightarrow(1,0,1)$ along $1=x^2+y^2$ for $z=1$

  • $(1,0,1)\rightarrow(0,0,0)$ along $z=x^2$ for $y=0$

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  • $\begingroup$ But I need to use stoke theorem to evaluate it. I got my answer 0 when curlF dot product with n. Is it correct the answer is 0? $\endgroup$ – Garett Sep 21 '13 at 8:05
  • $\begingroup$ If you use Stokes theorem, there would be no curf F $\endgroup$ – Shuchang Sep 21 '13 at 8:41
  • $\begingroup$ But i got curl F=$(-z^2)j$ $\endgroup$ – Garett Sep 21 '13 at 9:11
  • $\begingroup$ That's correct. $\endgroup$ – Shuchang Sep 21 '13 at 9:19
  • $\begingroup$ so the answer for this question is 0 is it? $\endgroup$ – Garett Sep 21 '13 at 9:26

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