Evaluate $\iint z \, dS$ where $S$ is the upper hemisphere of radius $2.$
I worked out $r_x = \langle -2\sin(x)\sin(y),2\cos(x)\sin(y),0\rangle$ and $r_y = \langle 2\cos(x)\cos(y),2\sin(x)\cos(y),-2\sin(y)\rangle$ .
Now taking the cross product of this then taking the magnitude of the cross product will take a very long time so is there an easier way of doing this considering they havent given any point. I read somewhere that
$$\left \| \frac{\partial r}{\partial x}\times \frac{\partial r}{\partial y} \right \| = \left \| \frac{\partial r}{\partial x} \right \|\left \| \frac{\partial r}{\partial y} \right \|$$
Is this true under any circumstance?