Evaluate $${\int \int}_S \underline{F} \cdot d \underline{a}$$
$$\underline{F} = (y+z, x+z, y+x)$$
$S$ is the portion of the surface of the cube bounded by the planes $x=0, y=0, z=0, x=1,y=1,z=1$ with outward normal.
I have a solution where it seems like they parametrised the 6 squares of the cube. I just don't understand how we would get these parametrisations?
E.g.
$$S_1^{-}: \underline r_1^- = (u,1,v)^{T}$$
$$S_2: \underline r_2 = (u,v,1)^{T}$$
The first of these is the square where $y=1$; it is parametrized by $0\le u, v\le 1$ (where $u$ and $v$ represent the $x$ and $z$ coordinates). The second is the square where $z=1$, parametrized by $0\le u, v \le 1$ (where $u$ and $v$ represent the $x$ and $y$ coordinates). Have you tried drawing a picture and thinking about what a general point on each face looks like?