Converting to cylindrical and then taking div and curl

[EngageEngage]

Homework Statement

Change to cylindrical coordinates and find the divergence

F = <x, y, 0>/(x^2 + y^2)

Homework Equations

\nabla . F = \frac{1}{\rho}\frac{\partial\rho F}{\partial\rho}+\frac{1}{\rho}\frac{\partial F}{\partial\theta}+\frac{\partial F}{\partial z}

The Attempt at a Solution

i changed into cylindrical by simply having x = \rhocos\theta
y =\rhosin\theta and x^2 +y^2 = \rho^2. THen, using the above formula I get cos(theta)/rho^2. The answer is 0 however. Can someone please tell me what i could have messed up?

 

[EngageEngage]

Wait, must I also convert the i,j,k unit vectors to e(rho), e(theta), and ez before I can use vector operations? If i do this i get the right answer, but I'm still not sure -- my book doesn't have a single example excersise so I don't have too much to work off of. If someone could help me out that would be appreciated greatly. Thank you

 

[Dick]

Wait, must I also convert the i,j,k unit vectors to e(rho), e(theta), and ez before I can use vector operations? If i do this i get the right answer, but I'm still not sure -- my book doesn't have a single example excersise so I don't have too much to work off of. If someone could help me out that would be appreciated greatly. Thank you

Sure. You have to convert the basis vectors as well. You should get F_r=1/r and the other components zero, so div(F)=0.

 

[EngageEngage]

that is in fact what I got. Thank you very much for the help -- unfortunately my book offers only theory and no examples.