Let $\vec{V} = (5x^2, -3xy^2 , z)$ calculate the flux along the path of the rectangle on $z=0$, $1\le x \le2$,$ 2\le y \le5 $
Since i need to calculate the flux along a closed route i will use Green's Theorem to calculate the flux -
$\int_2^5 dy \int_1^2 dx \Big(\dfrac{-3xy^2}{\partial x} - \dfrac{5x^2}{\partial y}\Big)$ = $\int_2^5 (-3y^2)dy \int_1^2 dx$ = $[-y^3]^5_2 = -117$
Calculate the flux of $\vec{V} = (x^2+z^3, 2xyz^3 , xz^4)$ through the closure of the cuboid along $(\pm 1, \pm 2, \pm3)$
I will use Gauss's Theorem
$\int\int\int_V \vec{\nabla} \cdot \vec{F}dv = \int\int\int_V (3x+2xz^3)dxdydz = \int_{-3}^3 dz\int_{-2}^2 dy\int_{-1}^1 (3x+5xz^3)$
So if i am right, the flux is $0$ since the function is an odd function ?
Thank you very much !