Divergence theorem and surface integrals

[henryc09]

Homework Statement

Consider the following vector field in cylindrical polar components:
F(r) = rz^2 r^{^} + rz^2 theta^{^}
By directly solving a surface integral, evaluate the flux of F across a cylinder
of radius R, height h, centred on the z axis, and with basis lying on the
z = 0 plane.
Using the formula derived in the previous question, evaluate the divergence
∇ · F. Verify the divergence theorem by integrating ∇ · F over the volume
of the cylinder above.

Homework Equations

The Attempt at a Solution

I think I can do the second part (I can definitely work out the divergence of F) although I'm not 100% on that. I really don't know how to go about doing the surface integral. Do you need to do the integral over each side of the cylinder? I'm not really sure how to go about doing it over any side though to be honest. Any help would be appreciated.

 

[mighty2000]

Hello,

First, let's define the surface of the cylinder in cylindrical polar coordinates. Since the cylinder is centered on the z axis and has a basis on the z = 0 plane, we can define the surface as follows:

r = R, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ h

To evaluate the flux of F across this surface, we can use the surface integral formula:

∫∫S F · dS = ∫∫S F(r) · r dθ dz

where dS is the surface element and r is the unit vector in the direction of the surface normal. In this case, r = r^, since the surface is perpendicular to the z axis.

Plugging in the given vector field, we get:

∫∫S F · dS = ∫∫S (rz^2 r^ + rz^2 θ^) · r dθ dz
= ∫∫S (r^2z^2 + r^2z^2) dθ dz
= ∫∫S 2r^2z^2 dθ dz

Now, we can evaluate this integral over each side of the cylinder separately. Let's start with the top and bottom surfaces, where θ ranges from 0 to 2π and z ranges from 0 to h. This gives us:

∫∫S 2r^2z^2 dθ dz = 2∫∫S r^2z^2 dθ dz
= 2∫∫S R^2z^2 dθ dz
= 2R^2∫∫S z^2 dθ dz
= 2R^2 ∫0h ∫02π z^2 dθ dz
= 2R^2 ∫0h 2πz^2 dz
= 4πR^2 ∫0h z^2 dz
= 4πR^2 (h^3/3)
= 4πR^2h^3/3

Now, for the curved surface of the cylinder, we have θ ranging from 0 to 2π and z ranging from 0 to h. However, the curved surface has a different surface element, which can be expressed as dS = R dθ dz. This gives us: