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Riemann Manifold is not rectangular or spherical

  1. Aug 13, 2013 #1
    I want to be able to formulate [tex]x^{n}[/tex] coordinate system.
    [tex]x^{n} =(x^{1}, x^{2}, x^{3}, x^{4}) [/tex]
    How do you do this when the Riemann Manifold is not rectangular or spherical?
    Also how do you differentiate with respect to "s" in that case.
    [tex]\frac{dx^n}{ds}[/tex]
     
  2. jcsd
  3. Aug 13, 2013 #2
    you can not simply do ##\frac{dx^n}{ds}##, because you do not have a parameter ##s##.

    You should do something like this.
    Make up a curve which is parameterized by ##s##.
    ##s## is your parameter along the curve.

    Now you have

    ##x^{n}(s) = (x^{1}(s), x^{2}(s), x^{3}(s), x^{4}(s))##

    And now you can do ##\frac{dx^n}{ds}## just fine.
    Which is your tangent vector to the curve (might not be unit length).

    Make your own coordinates.
    For a 2d-surface you could use ##u## and ##v## as coordinates.

    What do you want to do?
     
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