Multivaraible Calculus. surfaces in R2 andR3

ScullyX51

1. The problem statement, all variables and given/known data
In class, we studied functions ⃗r : I → R^3, where I ⊂ R is some interval.
Let us now consider a function
⃗r : U → R^3, U ⊂ R^2
That is, we have a function, ⃗r, which sends a point (u, v) in the plane to a
point (vector) in R3 . You may call it a “vector function of two arguments”
or a “vector function of a vector argument”. To be specific:
Let
U = {(u, v), v > 0} ⊂ R^2
⃗r(u, v) = (√v/2 cosh u, √v sinh u, v)
or, equivalently,
x = √v/2 cosh u
y = √v sinh u
z = v

1.Verify that the points ⃗r(u, v) satisfy the equation z = 4x^2− y^2
.
2.Identify the surface S given by the equation z = 4x^2-y^2
Is it an ellipsoid, paraboloid, hyperboloid (and which one), cone, cyllinder?



2. Relevant equations



3. The attempt at a solution
I know that the surface given by the equation is a hyperbolic parabloid, but I have no idea how to approach showing that the points satify the equation. When I tried just plugging the values of x,y, and z into the equation I end up with something very messy, and I'm not quite sure what I am supposed to be solving for. If anyone could explain this to me it would be great!!
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution