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**arbolis** Let $\displaystyle F(x,y,z)=x \vec i + y \vec j + yz \vec k$.

Calculate the flux of the curl of $\displaystyle F$ through the surface of the semi-sphere $\displaystyle x^2+y^2+z^2=4$, $\displaystyle z\leq 0$, oriented positively. Use Stokes theorem. Then use another way that does not involve Stokes' theorem to solve the problem.

My attempt : $\displaystyle \iint_ S curl F \hat n dS= \int _{\partial S} F dS$.

$\displaystyle curl F = z \vec i$.

$\displaystyle n=\frac{-x}{\sqrt{4-x^2-y^2}}$.

Hence $\displaystyle \iint_ S curl F \hat n dS=\iint _S \frac{-xz}{\sqrt{4-x^2-y^2}}dA= \iint _S -xdA=-\int _0^{2\pi} \int_ 0^2 r^2 \cos (\theta) drd\theta=0$. How is that possible? The field is not conservative and the surface is closed, hence it is impossible that the net flux passing through the semi-sphere give $\displaystyle 0$.

I want to tackle it by calculating $\displaystyle \int _{\partial S} F dS$ but with no success. I parametrize $\displaystyle \partial S$ : $\displaystyle r(t)=(2\cos t , 2\sin t, 0)$. Thus $\displaystyle r'(t)=(-2\sin t, 2 \cos t , 0)$.

Now I don't know what to do, is it a line integral of a scalar field or vector field? Seems more like a vector field to me, let's try.

$\displaystyle \int _0^{2\pi} -4 \cos (t) \sin (t)+4 \cos (t) \sin (t)+0 dt =0$. What's wrong?!

Using the divergence theorem I got $\displaystyle -\frac{64 \pi ^3}{3}$. However I do not know where to check if I didn't made a mistake of sign, since they said the surface to be oriented positively. The divergence is worth $\displaystyle 2+y$ so it can be positive or negative.

I'm all confused. What's the difference between Stokes and Gauss' theorems? They can both calculate the flux through a surface, hence they're equivalent?!