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Coordinate Transformation (and using line elements)

  1. Jan 21, 2009 #1
    This is from Hartle's GR book, in one of the first chapters it talks about diff geom, nothing too advanced, but I am learning on my own.

    1. The problem statement, all variables and given/known data
    It's part E I have trouble with. Read e. and skip to last para if you want.

    Consider this coordinate transformation:

    x=uv , y=(u^2 - v^2)/2

    a. Sketch curves of constant u and v in xy plane

    b. Transform the line element ds^2=dx^2 + dy^2 into u,v coordinates

    c. do curves of constant u and v intersect at right angles?

    d. find the equation of a circle of radius r centered at the origin in terms of u and v

    e. Calculate the ratio of the circumference to the diameter of a circle using uv coordinates

    2. Relevant equations

    The line element in rectangular coordinates, but that's written above

    3. The attempt at a solution
    Okay, I think curves of constant u and v are vertical parabolas, one facing up and one facing down.

    For the line element, I found ds^2= (u^2 + v^2)du^2 + (u^2 + v^2)dv^2

    I don't really know how to do c. But that doesn't bother me much,

    For the equation of a circle, I got (u^4)/2 + (v^4)/2 = R^2

    BUT it's e. I cannot do. I think it is the crux of the question, and I think it's vital that I master this concept. Would it be some sort of line integral using ds over the diameter, and the circumference of the circle? I'm not sure I know how to formulate it correctly. Also not sure what the equation of a straight line in uv coordinates is... Is it v=0?
     
  2. jcsd
  3. Jan 22, 2009 #2

    tiny-tim

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    Hi darkSun! :smile:
    Nooo … you multiplied your 2's wrong … try again! :wink:
     
  4. Jan 30, 2009 #3
    Oh, I see the error.

    So the equation of the circle is (u^4)/4 + (v^4)/4 +((uv)^2)/2 = R^2

    But still, how are the line integrals set up? Is it just a matter of setting up the integral in rectangular coordinates then switching to uv coordinates and substituting the Jacobian?

    Or is there a way to do it directly from uv coordinates?

    Also, for my curiosity, is this what a parabolic coordinate system looks like?
     
  5. Jan 31, 2009 #4

    tiny-tim

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    = (u2 + v2)2/4 :wink:
     
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