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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.
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