Consider the vertical vector field $\mathbf{u}=-\mathbf{k}$ representing a constant downward flow of rain on a sloped roof $S:z = 4-2x-y$ where $x, y, z\ge0$. Find the velocity flux in the negative $z$-direction through the roof. So first I shall parametrise the surface $S$:
I have parametrised the surface as $\mathbf{r}(u, v) = (u, v, 4-2u-v)$ where $0\le u\le 2$ and $0\le v\le 4-2u$. A normal to this surface is $\mathbf{n} = -2\mathbf{i}+\mathbf{j}+\mathbf{k}$. Therefore the integral is \begin{align}\iint_{S} \mathbf{u}\cdot \mathbf{dS} & = \int^{2}_{0}\int^{4-2u}_{0}(-\mathbf{k})\cdot(-2\mathbf{i}+\mathbf{j}+\mathbf{k}) \ {\rm d}v \ {\rm d}u\\&= -\int^{2}_{0}\int_{0}^{4-2u} {\rm d}v \ {\rm d}u\\&=-\int^{2}_{0}4-2u \ {\rm d}u\\&=-\Big[4u-u^{2}\Big]^{2}_{0}\\&= -(8-4) = -4.\end{align}
But the answer is $4$, what have I done wrong?