Question

Using Gauss’ Theorem calculate the flux of the vector field F = x î + y j + z k r through
the surface of a cylinder of radius A and height H, which has its axis along the z-axis
and the base of the cylinder is on the xy-plane.

Answer 1

$$F=(x,y,z) \Rightarrow \nabla F=\partial_x x+\partial_y y+\partial_z z=3$$

Let $C$ is our cylinder and $S$ is it's surace. By Gauss Theorem

$$\iint\limits_S F \, dS=\iiint\limits_C \nabla F \, dV=3 \iiint\limits_C \, dV=3\pi A^2 H$$