Div, grad and curl in cylindrical polar coordinates

[maggie56]

Homework Statement

Hi,
i am trying to find the div, grad and curl in cylindrical polar coordinates for the scalar field
$$\ phi = U(R+a^2/R)cos(theta) + k*theta$$ for cylindrical polar coordinates (R,theta,z)
I have attempted all three and would really appreciate it if someone could tell me if the answers look ok as I am really not sure whether i have correctly followed the method
Thank you
Sorry i forgot to put that its the curl of the gradient and divergence of the gradient that I am finding. I guess i have a non zero answer for curl of gradient because U,a and k are constants so my answer would be zero for certain U,a,k.

Homework Equations

$$\ phi = U(R+a^2/R)cos(theta) + k*theta$$ U,a,k constants

The Attempt at a Solution

For gradient of phi $$\ U(1-a^2/R^2)cos(theta)$$ R'hat' - $$\[ U(1+a^2/R^2)sin(theta) + k/R]$$
theta'hat'

Curl of phi $$\ sin(theta)(2Ua^2/R^4 + U/R - a^2/R^3) - k/R^3$$ z'hat'



divergence of phi is zero

 

[vela]

The gradient is correct, but the curl and divergence aren't. You can't take the curl and divergence of a scalar field.

 

[maggie56]

Sorry its the curl of the gradient and the divergence of the gradient.
i know that the curl of the gradient is always zero?

 

[vela]

Right, the curl is 0. Mathematica gave me a different result for the divergence, though. (I think you swapped "curl" and "divergence" in the original post. Or maybe not. Either way, they're both incorrect.)

 

[maggie56]

I have looked at the curl and divergence again. u is the gradient of phi.

u = $$\ U(1-a^2/R^2)cos(theta)$$ R'hat' - $$\[ U(1+a^2/R^2)sin(theta) + k/R]$$ theta'hat'

so curl of u is
$$\ [U/R -a^2U/R^3 - 2a^2U/R^4)sin(theta) - k/R^3$$ z'hat'

divergence of u is 0

Do these answers look better? since U,a and k are constants i have an expression for the curl of the gradient but this could be zero for certain U,a and k.

 

[vela]

No, the curl of the gradient is 0 for all U, a, and k, and the divergence is not identically 0.

Show your work.

 

[maggie56]

my working
for the curl of the gradient

$$\ 1/R [ 0<b>R'hat'</b> + 0 <b>theta'hat'</b> + -2Ua^2/R^3 sin(theta) - k/R^2 + U(1-a^2/R^2 sin(theta) ]$$

= $$\ sin(theta)(-2Ua^2/R^4 + U/R - a^2/R^3) - k/R^3$$

 

[maggie56]

my matrix for the curl is

R'hat' Theta'hat' Z'hat on top line
d/dR d/dtheta d/dz on middle line
$$\ U(1-a^2/R^2)cos(theta)$$ $$\ -U(1+a^2/R^2)sin(theta)+k/R$$ 0 on bottom line

with a 1/R on the outside
the -U is in the middle column and zero is for z column

 

[vela]

That's not the correct method. Take a look here:

http://hyperphysics.phy-astr.gsu.edu/hbase/curl.html