How many cubic metres of fluid cross the upper hemisphere $x^2+y^2+z^2=1$, $z\ge0$ per second if the velocity of the flow is $\mathbf{u}=\mathbf{i}+x\mathbf{j}+z\mathbf{k}$ metres per second.
So I have parametrised the hemisphere:
$$\mathbf{r}(\theta, \phi) = (\sin{\theta}\cos{\phi}, \sin{\theta}\sin{\phi}, \cos{\theta})$$ where $0\le\theta\le \pi/2$, $0\le\phi\le2\pi$. Now I want to find $$\int_S \mathbf{u} \cdot \mathbf{dS}$$
Now I also know $\mathbf{dS}=r^{2}\sin{\theta}{\rm d}\theta{\rm d}\phi \mathbf{\hat{r}}$. How do I continue?
Form the inner product of $\hat u$ with the unit normal. Use
$$\begin{align} \hat x\cdot \hat r&=\sin(\theta)\cos(\phi)\\\\ \hat y\cdot \hat r&=\sin(\theta)\sin(\phi)\\\\ \hat z\cdot \hat r&=\cos(\theta) \end{align}$$
along with $u_y=x=\sin(\theta)\cos(\phi)$ and $u_z=z=\cos(\theta)$ on the unit circle.
Then, proceed to integrate three terms
$$\begin{align} &\int_0^{2\pi}\int_0^{\pi/2} \sin(\theta)\cos(\phi)\,\sin(\theta)\,d\theta\,d\phi \tag 1\\\\ &\int_0^{2\pi}\int_0^{\pi/2} \sin(\theta)\cos(\phi)\sin(\theta)\sin(\phi)\,\sin(\theta)\,d\theta\,d\phi \tag 2\\\\ &\int_0^{2\pi}\int_0^{\pi/2} \cos(\theta)\,\cos(\theta)\,\sin(\theta)\,d\theta\,d\phi \tag 3 \end{align}$$
All three integrals should be straightforward to evaluate. The first two integrate to zero since the integrals from $0$ to $2\pi$, with respect to $\phi$, of $\cos\phi$ and $\cos(\phi)\sin(\phi)$ are obviously zero.