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Hyperbolic matrix transform

  1. Sep 15, 2006 #1
    I am not really sure if I am doing this problem correctly if you could point out any errors that would be great.

    The problem: The coordinates of a hyperbolic system (u,v,z) are related to a set of cartesian coordinates (x,y,z) by the equations

    u=x^2-y^2
    v=2xy
    z=z
    Determine the transformation matrix [a] that takes the cartesian componets of a vector to the hyperbolic components.

    What I did:
    the transformation matrix is given by a_ij = dx'_ij/dx_ij, where dx'/dx are partial derivatives and x' corresponds to u,v,z.

    giving a matrix of | 2x -2y 0 | | x -y 0|
    | y x 0 | = |y x 0|
    |0 0 1 | |0 0 1 |
    After dividing first two rows by 2.

    I know that [a][a]^T = [1]
    for [a][a]^T = | ( x^2+y^2) 0 0 |
    | |
    | 0 (x^2+y^2) 0|
    | |
    | 0 0 1|

    which can only equal the identity if x^2+y^2=1
    I was wandering if this looks ok
    thanks
     
  2. jcsd
  3. Sep 15, 2006 #2
    sorry [a][a]^T is a little hard to read. My comp. for some reason doesnt like to work with latex.

    for [a][a]^T i got

    (x^2+y^2) 0 0
    0 (x^2+y^2) 0
    0 0 1
     
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