Question

Using Gauss’ Theorem calculate the flux of the vector field F = x i^ + y j^ + z k^ 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

The Gauss theorem states that the flux $\oint_{\partial V} \left( \boldsymbol{F} \cdot \boldsymbol{n} \right) dS$ of a vector field $\boldsymbol{F}$ over a boundary $\partial V$ of volume $V$ is equal to the volume integral $\int_V \left( \nabla \cdot \boldsymbol{F} \right) d V$. Calculating the divergence of our vector field, we have $\nabla \cdot \boldsymbol{F} = 3$, and the volume integral is

$$\int_V \left( \nabla \cdot \boldsymbol{F} \right) d V = 3 \int_V d V = 3 V = 3 \pi A^2 H \, ,$$

where we have taken into account that $V = \pi A^2 H$.

Answer: $3 \pi A^2 H$.