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Could anyone help me with this question?

I need to compute the flux of vector field $\textrm{curl } F$ through the hemisphere

$$x = \sqrt{1 - y^2 - z^2}$$

Positively oriented, with the vector field

$$F(x,y,z) = \langle e^{xy}cosz, x^2z,xy \rangle$$

I've already tried to do spherical coordinates but I don't know if it is the way, because the terms became to large and not seems to take me anywhere.

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Use Stokes Theorem:

$$ \int_S \nabla \times \vec{F} \; \mathrm{d}\vec{S} = \oint_{\partial S} \vec{F} \; \mathrm{d}\vec{r}$$

where the line integral is along the open boundary of the hemisphere - which for your geometry is just the unit circle in the yz-plane at x = 0; from here the computation should be straightforward.

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