# Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k$

Compute the flux of the vector field $F(x,y,z) = \left(2x-y^2\right) \mathbf i +\left( 2x - 2yz\right) \mathbf j + z^2 \mathbf k$ through the surface consisting of the side and bottom of the cylinder of radius two and height two, i.e., $\{(x,y,z)\vert x^2+y^2 =4, 0 \le z \le 2\}$.(Note that this surface does not include the top of the cylinder.)

That is, compute the surface integral $$\iint_{\mathbf S} \mathbf F \cdot \mathbf n dS$$ where $\mathbf F$ is the vector field above, S is the bottom and side (but not the top) of the cylinder above, and $\mathbf n$ is the outward pointing unit normal vector to thesurface.

• Do you have an idea of where to start the parameterization of the surface? – user217285 Mar 16 '16 at 1:42
• Hint: $\nabla \cdot \vec{F}=2$. – Kuifje Mar 18 '16 at 3:10

$$\displaystyle \int_{z=0}^{2}\iint_{x^2+y^2\leq 4} 2dxdydz=16\pi$$