Answer to Question #87975 in Calculus for ajay

Question

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

Gauss' theorem states

$$Flux=\int\overrightarrow{F}d\overrightarrow{A}=\int div\overrightarrow{F}dV$$

$$div\overrightarrow{F}={\partial F_x \over \partial x}+{\partial F_y \over \partial y}+{\partial F_z \over \partial z}$$

$$\overrightarrow{F}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$$

$$div\overrightarrow{F}={\partial x \over \partial x}+{\partial y \over \partial y}+{\partial z \over \partial z}=1+1+1=3$$

Hence

$$Flux=\int\overrightarrow{F}d\overrightarrow{A}=\int div\overrightarrow{F}dV=\int 3 dV=3V=3\pi A^2H$$