Is the transformation matrix correct?

[matpo39]

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

 

[matpo39]

sorry [a][a]^T is a little hard to read. My comp. for some reason doesn't 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