Parabolic Coordinates: u,v,φ in x,y,z

[latentcorpse]

Show that the parabolic coordinates $(u,v,\phi)$ defined by

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

now I am 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 $\vec{e_u}$

$u^2v^2=x^2+y^2$ and $u^2-2z=v^2$
$u^2(u^2-2z)=x^2+y^2 \Rightarrow u^4-2u^2z=x^2+y^2$ - i.e. my problem is I am finding it impossible to rearrange for u...

 

[CompuChip]

You didn't finish the question ("Show that the parabolic coordinates what?").
If you want to solve
$$u^4 - 2u^2z = x^2 + y^2$$
you could set U = u² and solve the quadratic equation
$$a U^2 + b U + c = 0$$
with a = 1, b = - 2 z, c = -(x^2 + y^2); for U.

 

[latentcorpse]

yep. that's my bad. i need to show they're orthogonal.