Here's the question:
Evaluate $\iint_{S} \boldsymbol{F} \cdot \boldsymbol{\hat{n}}$ if
$\boldsymbol{F} = (x+y) \boldsymbol{\hat{i}} + x \boldsymbol{\hat{j}}
+z \boldsymbol{\hat{k}}$ and $S$ is the surface of the cube bounded by the planes $x=0$,$x=1$,$y=0$, $y=1$, $z=0$ and $z=1$.
Here's my attempt:
Suppose the faces whose equations are $x=0$,$x=1$,$y=0$, $y=1$, $z=0$ and $z=1$ are respectively named $S_1$, $S_2$ and so on respectively and let $\boldsymbol{\hat{n}}$ denote the unit vector normal to them.
Now on $S_1$, $\boldsymbol{F} = y \boldsymbol{\hat{j}}
+z \boldsymbol{\hat{k}}$, $\boldsymbol{\hat{n}}=\boldsymbol{\hat{i}}$. Therefore $\iint_{S_1} \boldsymbol{F} \cdot \boldsymbol{\hat{n}} = \int_{0}^{1} \int_{0}^{1} y \mathrm{d}y \mathrm{d}z = \frac{1}{2}$.
Similarly we have $\iint_{S_2} \boldsymbol{F} \cdot \boldsymbol{\hat{n}} = \frac{3}{2}$, $\iint_{S_3} \boldsymbol{F} \cdot \boldsymbol{\hat{n}} = \frac{1}{2}$, $\iint_{S_4} \boldsymbol{F} \cdot \boldsymbol{\hat{n}} = \frac{1}{2}$, $\iint_{S_5} \boldsymbol{F} \cdot \boldsymbol{\hat{n}} = 0$ and $\iint_{S_6} \boldsymbol{F} \cdot \boldsymbol{\hat{n}} = 1$.
Hence overall we have $\iint_{S} \boldsymbol{F} \cdot \boldsymbol{\hat{n}} = 4$. But the answer on the textbook seems to be $2$. I checked everything up and there doesn't seem to be any error on my part but I was wondering how the answer doesn't match up.
Best Answer
The problem, is that you are not using outward pointing normal vectors. Every plane has two unit normal vectors, but only one of them is pointing outward. For a flux integral, we use the convention that the flux is the flow out of an object.
In your case, the outward pointing normal vector for $S_1$ is $\langle -1,0,0\rangle$, which changes the sign of your answer. The outward pointing normal vector for $S_2$ remains $\langle 1,0,0\rangle$, so that answer doesn't change. There are two more vectors which need to swap signs, and after that, you'll get $$ \left(-\frac{1}{2}\right)+\frac{3}{2}+\left(-\frac{1}{2}\right)+\frac{1}{2}+(-0)+1=2. $$