Stokes Theorem Integral

Question

Evaluate $\iint_S \langle F,\eta\rangle \,d\sigma$ where $F(x,y,z)=(xz,yz,z^2)$ and $S$ is the upper hemisphere of radius $1$ centred at the origin.

$\eta$ is the unit vector perpendicular to the unit tangent vector.

Answer 1

Use spherical coordinates.

$$x=\sin{\theta} \cos{\phi}$$ $$y=\sin{\theta}\sin{\phi}$$ $$z=\cos{\theta}$$

You can then show that, as $\vec{\eta}$ is just the radial vector $(x,y,z)$:

$$\vec{F}\cdot \vec{\eta} = \cos{\theta}$$

and the surface integral is

$$\int_0^{\pi/2} d\theta \, \sin{\theta} \, \cos{\theta} \: \int_0^{2 \pi} d\phi = \pi$$

Answer 2

If the intention is to use Stokes's Theorem (or, in this case, the Divergence Theorem), attach the unit disk at the bottom and deduce that the flux of $\vec F$ across $S$ is equal to sum of the flux upwards across the disk at the bottom and $\iiint_V \text{div} \vec F\,dV$.