Question

Using Gauss’s theorem, calculate the flux of the vector field F =xˆi + y ˆj + zkˆ
through the surface of a cylinder of radius A and height H, which has its axis
along the z-axis. The base of the cylinder is on the xy plane.

Answer 1

For a vector field

$$\textbf{F}=P\textbf{i}+Q\textbf{j}+R\textbf{k}=x\textbf{i}+y\textbf{j}+z\textbf{k},$$

the flux through the 3-dimensional surface element $S$ is

$$\Phi=\iint\textbf{F}\text{ d}S=\iiint\text{ div}\textbf{F}\text{ d}V,\\ \space\\ \text{ div}\textbf{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}=\\ \space\\ =\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=\\ \space\\ =1+1+1=3.\\ \space\\ \iiint\text{ div}\textbf{F}\text{ d}V=\iiint3\text{d}V=3V=3\pi A^2H.$$