2019-03-12T07:23:22-04:00
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
2019-03-15T12:47:51-0400
Gauss' theorem states
F l u x = ∫ F ⃗ d A ⃗ = ∫ d i v F ⃗ d V \rm{Flux}=\int \vec F d\vec A=\intop \rm{div}\vec F dV Flux = ∫ F d A = ∫ div F dV Since
d i v F ⃗ = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z = ∂ x ∂ x + ∂ y ∂ y + ∂ z ∂ z = 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 div F = ∂ x ∂ F x + ∂ y ∂ F y + ∂ z ∂ F z = ∂ x ∂ x + ∂ y ∂ y + ∂ z ∂ z = 3 we obtain
F l u x = ∫ d i v F ⃗ d V = ∫ 3 d V = 3 V = 3 π A 2 H \rm{Flux}=\intop \rm{div}\vec F dV=\intop 3 dV=3V=3 \pi A^2 H Flux = ∫ div F dV = ∫ 3dV = 3V = 3 π A 2 H
Need a fast expert's response?
Submit order
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS !
Comments