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Coordinate systems for divergence

  1. Mar 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Compute 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)
    ∂F_y/∂y = (∂F_y/∂rho)(∂rho/∂y) + (∂F_y/∂theta)(∂theta/∂y) + (∂F_y/∂z)(∂z/∂y)
    ∂F_z/∂z = (∂F_z/∂rho)(∂rho/∂z) + (∂F_z/∂theta)(∂theta/∂z) + (∂F_z/∂z)(∂z/∂z)

    ∂rho/∂x = x/∂ = costheta
    ∂theta/∂x = -y/rho^2 = -sintheta/rho
    ∂z/∂x = 0
    ∂rho/∂y = y/∂ = sintheta
    etc. (these are the transformational equations)

    Then I try inputing this into 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
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