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Vector calc question - coordinate systems

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant 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?
     
  2. jcsd
  3. Mar 22, 2009 #2

    gabbagabbahey

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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?
     
  4. Mar 22, 2009 #3
    I don't know. I can't find the relation anywhere in my book. What is it?
     
  5. Mar 22, 2009 #4

    gabbagabbahey

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    You'll have to derive it....I'll give you a hint: [tex]F_x=\vec{F}\cdot\hat{i}[/tex]....
     
  6. Mar 22, 2009 #5
    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.
     
  7. Mar 22, 2009 #6

    gabbagabbahey

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    Right...



    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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