Divergence and curl of spherical polar coordinates

[maggie56]

Homework Statement

Hi,
i am trying to find the div and curl in spherical polar coordinates for the vector field, F

I have attempted both 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

Homework Equations

F [tex] \= -2rcos(theta) r'hat'+ 3rsin(theta) theta'hat' [/tex]

The Attempt at a Solution

divergence of F i have

$$\ 6*cos^2(theta)/sin(theta) - 6*cos(theta)$$
it should be 6 cos^2 in the first term above, i can't get it to show for some reason


Curl of F i have

4sin(theta) phi'hat'

 

[mighty2000]

- 6cos(theta) r'hat'


Hi there,

Your attempt at finding the divergence and curl in spherical polar coordinates for the given vector field looks like it is on the right track. However, there are a few things that need to be corrected in your solutions.

Firstly, for the divergence, the correct expression should be:

div(F) = 3cos(theta) + 6cos(theta)/sin(theta)

This can be obtained by using the definition of divergence in spherical coordinates:

div(F) = (1/r^2)*(∂(r^2F_r)/∂r + ∂(rF_theta)/∂θ + (1/sin(theta))*∂(F_phi)/∂phi)

where F_r, F_theta and F_phi are the components of the vector field F in spherical coordinates.

For the curl, the correct expression should be:

curl(F) = (1/r)*(-3sin(theta) phi'hat' + 6cos(theta) r'hat')

Again, this can be obtained using the definition of curl in spherical coordinates:

curl(F) = (1/r)*(1/sin(theta))*∂(rF_phi)/∂theta - ∂(F_theta)/∂phi + ∂(rF_theta)/∂r - (1/sin(theta))*∂(F_r)/∂phi

I hope this helps clarify any doubts you may have had in your solutions. Keep up the good work!