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Vector calc question - coordinate systems

  • Thread starter jaejoon89
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1. Homework Statement

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

2. 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)

3. 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

Homework Helper
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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?
 
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

Homework Helper
Gold Member
5,001
6
You'll have to derive it....I'll give you a hint: [tex]F_x=\vec{F}\cdot\hat{i}[/tex]....
 
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

Homework Helper
Gold Member
5,001
6
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
 

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