Let

γ : R2 → R3 be a unit speed curve of nonvanishing curvature and let its scalar curvature, torsion and Frenet frame be denoted κ, τ and [u, n, b], as usual. Consider the mapping M : R2 → R3, M(s, t) = γ(s) + tb(s).

1)Show that M is a regularly parametrized surface. (You may assume that M is one-to-one.)

You may assume the frenet formulae.

I got (u(s)-tτn(s)) x b(s) for the cross product of the partial derivatives (which I need to show is non-zero for all s,t), but how do I simplify further and is this correct?