Consider the surface patch σ(t,θ) = (coshtcosθ, coshtsinθ, t) where t is an element of the set of real numbers and θ is an element from (-pi, pi).

Show that σ defines a regular surface patch and find an orthonormal basis for the tangent space (TpS) at points of the form P = (cosht, 0, t)

I have done the regular surface patch part. Now just wondering how I go about the orthonormal basis part.

By differentiation:

u1 = ∂σ/dt = (sinhtcosθ, sinhtsinθ, 1) and

u2 = ∂σ/dθ = (-coshtsinθ, coshtcosθ, 1)

Using Gram-Schmidt, I found that u1 and u2 are already orthogonal since the inner product is 0.

I can then normalize u1 and u2:

v1 = u1/|u1| = (sinhtcosθ, sinhtsinθ, 1)/cosht

v2 = u2/|u2| = (-coshtsinθ, coshtcosθ, 1)/cosht

Now I don't know where to go from here...

I thought maybe I should check for a linear combination such that

(cosht, 0, t) = Av1 + Bv2

and then see if the constants A & B satisfy all 3 equations, but that didn't seem to work...