the way you worte it looks a little confusing.
so as far as I understind is:
⃗r : U → R3 ,U ⊂ R 2
We are given:
⃗r (u,v)= (√v/2 cosh u, √v2sinhu,v)
or the same as:
x= √v/2 cosh u
y- √vsinh u
The question is verify that the points r(u,v) satisfy the equation z= 4x^2-y^2. And identify what kind of surface z=4x^2-y^2 is.
This is what I did:
I identified the surface given by the equation z=4x^2-y^2 as a hyperbolic parabloid.
The part I am confused about is showing that it satisfies the equation. what I did is substituted the given values of x,y,and z into this formula, and I got:
I don't know where to go from here, would appreciate any suggestions.