Deriving the divergence in polar coordinates

[Amok]

Homework Statement

I want to find the divergence operator in polar coordinates (theta and r). I know how to write this operator in cartesian coordinates.

The Attempt at a Solution

I let F(F1,F2) be a vector field. I calculated the partial derivatives of F1 and F2 with respect to x and y respectively ( I said F1 and F2 were functions of theta and r and used the chain rule). That was easy, but I don't get the right answer. I'm pretty sure the reason I got it wrong has something to do with the fact that I didn't take into account the unit vectors in polar coordinates.

Could anyone show me how it's done or at least explain to me what I should do? I would've written more equations, but I don't really know how to do this. I hope it's clear.

 

[mighty2000]

Hello there! It sounds like you have the right idea, but as you mentioned, the unit vectors in polar coordinates need to be taken into account. In polar coordinates, the unit vectors are not constant, they vary with position. The unit vectors are given by e_r = cos(theta) i + sin(theta) j and e_theta = -sin(theta) i + cos(theta) j.

To find the divergence operator, you can use the following formula:

div(F) = (1/r)(d/dr)(rF_r) + (1/r)(d/dtheta)(F_theta), where F_r and F_theta are the components of the vector field F in the r and theta directions, respectively.

So, for example, if F = F_r e_r + F_theta e_theta, then F_r = F_r and F_theta = F_theta.

Hope this helps! Let me know if you have any further questions.