Calculating Flux of a surface

Question

I have to calculate the flux a surface but I don't really find the way to parametrize the surface. Moreover, I am not sure if I have to use Gauss theorem or Stoke's theorem. This is my exercise :

Let be $F = (xy + yz + zx, \cos(xyz), xy - xz + \cos(xyz) - 2x)$ a vector field.

$V = \{(x,y,z)| 0\le x, y, z \le 1\}$ (the cube) and $L$ is the surface intersection between the boundary of $V$ and the plane $y + z = 3/2$.

So, I have to calculate the integral over $L$ of $F$.

Thank you !

Answer 1

We need to evaluate

$$\iint_L \vec F \cdot \vec n \, dS$$

where

• $\vec n=\pm \frac1{\sqrt 2}(0,1,1)$ depending on the choosen orientation
• $\vec F \cdot \vec n =\pm \frac1{\sqrt 2}\left(2\cos (xyz)+xy-xz-2x\right)$ with $z=\frac32-y$
• $dS = \sqrt 2 \, dx\, dy$ with $0\le x\le 1$ and $\frac12\le y \le 1$