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Orthogonal Coordinates

  1. Apr 9, 2009 #1
    Show that the parabolic coordinates [itex](u,v,\phi)[/itex] defined by

    [itex]x=uv \cos{\phi} , y=uv \sin{\phi} , z=\frac{1}{2}(u^2-v^2)[/itex]

    now im a bit uneasy here because to do this i first need to find the basis vector right?

    so if i try and rearrange for u say and then normalise to 1 that will give me [itex]\vec{e_u}[/itex]

    [itex]u^2v^2=x^2+y^2[/itex] and [itex]u^2-2z=v^2[/itex]
    [itex]u^2(u^2-2z)=x^2+y^2 \Rightarrow u^4-2u^2z=x^2+y^2[/itex] - i.e. my problem is im finding it impossible to rearrange for u....
  2. jcsd
  3. Apr 10, 2009 #2


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    You didn't finish the question ("Show that the parabolic coordinates what?").
    If you want to solve
    [tex]u^4 - 2u^2z = x^2 + y^2[/tex]
    you could set U = u2 and solve the quadratic equation
    [tex]a U^2 + b U + c = 0[/tex]
    with a = 1, b = - 2 z, c = -(x^2 + y^2); for U.
  4. Apr 10, 2009 #3
    yep. that's my bad. i need to show they're orthogonal.
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