Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find infinitesimal displacement in any coordinate system

  1. Nov 15, 2015 #1
    I am wondering how can I find the infinitesimal displacement in any coordinate system. For example, in spherical coordinates we have the folow relations:
    [itex] x = \, \rho sin\theta cos\phi[/itex]
    [itex]y = \, \rho sin\theta sin\phi[/itex]
    [itex]z = \, \rho cos\theta [/itex]

    And we have that [tex]d\vec l = dr\hat r +rd\theta\hat \theta + r sin\theta d\phi \hat \phi [/tex]
    How do I found this for any system? What book has this explanation? I am not finding.
    Last edited by a moderator: Nov 15, 2015
  2. jcsd
  3. Nov 15, 2015 #2
    In general the infinitesimal displacement given in some coordinates [itex]q_i[/itex] is [itex]d\vec r=\frac{\partial \vec r}{\partial q_j}dq_j[/itex] (sum over j), so if you have a vector given in cartesian coordinates, and know how to transform those to the new coordinates, you can use the above formula. For more info, I'd suggest Riley & Hobson, chapters 10 and 26.

    Note: this works fine in Euclidean space [itex]\mathbb R^n[/itex], but for an arbitrary Riemannian manifold I'm not so sure.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Find infinitesimal displacement in any coordinate system