Question #86238
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.
1
Expert's answer
2019-03-15T12:47:51-0400

Gauss' theorem states

Flux=FdA=divFdV\rm{Flux}=\int \vec F d\vec A=\intop \rm{div}\vec F dV

Since

divF=Fxx+Fyy+Fzz=xx+yy+zz=3\rm{div}\vec F=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}=\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=3

we obtain

Flux=divFdV=3dV=3V=3πA2H\rm{Flux}=\intop \rm{div}\vec F dV=\intop 3 dV=3V=3 \pi A^2 H


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