(b) Show that there exist an open neighborhood U of p and an open neighborhood V of F (p) such that F : U → V is a bijection whose inverse is a C1-function

1. Quiz 9

(1) Suppose W is an open subset of R² and f : W → R be a C¹-function. Assume that p = (x0, y0) ∈ W and fy(p) 6= 0. Define a function F : W → R² by

F (x, y) = (x, f (x, y)).

(a) Compute DF (p) and show that the Jacobian J_F(p) = det DF (p) of F at p is nonzero.

(b) Show that there exist an open neighborhood U of p and an open neighborhood V of F (p) such that F : U → V is a bijection whose inverse is a C¹-function.

(c) Let G : V → U be the inverse of F : U → V. Let us write G(u, v) = (g(u, v), h(u, v)) for (u, v) ∈ V. Prove that

g(u, v) = u, f (u, h(u, v)) = v for all (u, v) ∈ V. Hint: F ◦ G = idV .

(d) Suppose f (p) = f (x₀, y₀) = c. Show that there exists δ > 0 and a C¹-function φ : (x₀− δ, x₀+ δ) → R

such that φ(x0) = y0 and f (x, φ(x)) = c for all x ∈ (x0− δ, x₀+ δ). Hint: you may choose U in Problem (1c) so that U is an open rectangle U = (x₀− δ, x₀+ δ) × (y₀− δ, y₀+ δ).

(2) Let W be an open subset of R³ and f : W → R be a C¹ function. Assume that p = (x₀, y₀, z₀) such that f_z(p) 6= 0. Define a function F : W → R³ by

F (x, y, z) = (x, y, f (x, y, z)).

(a) Show that there exist open neighborhood U of p and open neighborhood V of F (p) so that F : U → V is a bijection whose inverse G : V → U is a C¹-function.

(b) Write G(u, v, w) = (g(u, v, w), h(u, v, w), k(u, v, w)) for (u, v, w) ∈ V. Show that g(u, v, w) = u, h(u, v, w) = v, f (u, v, k(u, v, w)) = w

for (u, v, w) ∈ V.

(c) Assume that f (p) = c. Prove that there exists δ > 0 and φ : (x0− δ, x₀+ δ) × (y0− δ, y₀+ δ) → R so that

f (x, y, φ(x, y)) = c

for any (x, y) ∈ (x₀− δ, x₀+ δ) × (y₀− δ, y₀+ δ). Also show that f_x(x, y, φ(x, y)) + f_z(x, y, φ(x, y))φ_x(x, y) = 0

fy(x, y, φ(x, y)) + fz(x, y, φ(x, y))φy(x, y) = 0 for all (x, y) ∈ (x0− δ, x₀+ δ) × (y0− δ, y₀+ δ).

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