Vector calc question - coordinate systems

[jaejoon89]

Homework Statement

How do you derive the divergence in cylindrical coordinates by transforming the expression for divergence in cartestian coordinates?

Homework Equations

F = F_x i + F_y j + F_z k
div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z (divergence in Cartesian coordinates)

I need to transform this into

divF = (1/rho)(∂(rho*F_rho)/∂rho) + (1/rho)(∂F_theta/∂theta) + ∂F_z/∂z (divergence in cylindrical coordinates)

The Attempt at a Solution

Using the chain rule,
∂F_x/∂x = (∂F_x/∂rho)(∂rho/∂x) + (∂F_x/∂theta)(∂theta/∂x) + (∂F_x/∂z)(∂z/∂x)
Similarly for F_y and F_z

Then I rewrite the cartesian definition for divergence and obtain
divF = [(∂F_x/∂rho)costheta + (∂F_x/∂theta)(-sintheta/rho)] + [(∂F_y/∂rho)sintheta + (∂F_y/∂theta)(costheta/rho)] + ∂F_z/∂z

But how does that simplify to the expression in cylindrical coordinates?

 

[gabbagabbahey]

Stop creating multiple threads for the same problem.

As for your question; what are F_x,F_y and F_z in terms of F_rho, F_theta and F_z?

 

[jaejoon89]

As for your question; what are F_x,F_y and F_z in terms of F_rho, F_theta and F_z?

I don't know. I can't find the relation anywhere in my book. What is it?

 

[gabbagabbahey]

You'll have to derive it...I'll give you a hint: $$F_x=\vec{F}\cdot\hat{i}$$...

 

[jaejoon89]

Thanks for the suggestion. But I'm not sure I follow.

I know

F = F_p(p,theta,z)e_p + F_theta (p,theta,z)e_theta + F_z (p,theta,z)e_z

where p = rho

So does F_x = F_p(p,theta,z)e_p ? But divergence is not a vector so the e_p shouldn't matter... so I'm still not sure how to derive the relation. Again, thanks the help.

 

[gabbagabbahey]

Thanks for the suggestion. But I'm not sure I follow.

I know

F = F_p(p,theta,z)e_p + F_theta (p,theta,z)e_theta + F_z (p,theta,z)e_z

Right...

So does F_x = F_p(p,theta,z)e_p ?

No! F_x=F.i=( F_p(p,theta,z)e_p + F_theta (p,theta,z)e_theta + F_z (p,theta,z)e_z).i (the '.' means dot product)

Compute the dot product and then do the same for F_y and F_z