Suppose that S is a Cl surface with parametrization (, E) which is smooth at (x0, y0, z0) (x0, v0) Let (, I) be a parametrization of a C1 curve in E which passes through the point (u0, v0) i e , there is a to I such that (t0) (u0, v0) Prove that ( o ) '(t0)) (u u)(u0, v0) 0 ...

Let x y z u v and u v t Notice ...

Suppose that S is a Cl surface with parametrization (, E) which is smooth at (x0, y0,

Question:

Suppose that S is a Cl surface with parametrization (ϕ, E) which is smooth at (x0, y0, z0) = ϕ(x0, v0). Let (ψ, I) be a parametrization of a C1 curve in E which passes through the point (u0, v0) [i.e., there is a to ∈ I such that ψ(t0) = (u0, v0)]. Prove that (ϕ o ψ)'(t0)) ∙ (ϕu × ϕu)(u0, v0) = 0.