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Homework Help: The length element in cylindrical coordinates

  1. Sep 27, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that in cylindrical coordinates

    [itex]x = \rho cos \theta[/itex]
    [itex]y = \rho sin \theta[/itex]
    [itex]z = z[/itex]

    the length element ds is given by

    [itex]ds^{2} = dx^{2} + dy^{2} + dz^{2} = d \rho^{2} + \rho^{2} d \theta ^{2} + dz^{2}[/itex]

    2. Relevant equations


    3. The attempt at a solution

    Notice [itex]\rho = \sqrt{x^{2} + y^{2}}[/itex]

    I have found expressions for ∂x/∂θ, ∂x/∂ρ, ∂y/∂θ, ∂y/∂ρ, ∂ρ/∂x, ∂ρ/∂y, ∂θ/∂x, ∂θ/∂y, and of course it is trivial that dz = dz. Given all of these, how do i start using chain rule and total derivatives to find ds2? I guess because i have 8 partials but end up only ρ, θ, and z, i don't really know which ones to start with.

    Any help is greatly appreciated.
  2. jcsd
  3. Sep 27, 2013 #2


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    Homework Helper

    Start from
    dx = \frac{\partial x}{\partial \rho}d\rho + \frac{\partial x}{\partial \theta}d\theta
    Squaring both sides gives [itex]dx^2[/itex].
  4. Sep 27, 2013 #3
    yeah i got it. thanks.
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