Math205c Spring Term 2003

Assignment 21

1. Let f, g ∈ C^{1}(ℝ^{2}, ℝ) be given. Find necessary conditions for the existence of u ∈ C^{2}(ℝ^{2}, ℝ) such that

∇u = (f, g) .

Show that those conditions are also sufficient and derive a formula describing u in terms of f and g.

2. Let F ∈ C^{2}(D, ℝ^{n}) for some D ⊂^{o} ℝ^{n}. In order to solve F (x) = 0 one can use Newton iteration

x_{n+1} = x_{n} − DF (x_{n})^{−1}F (x_{n}) ,

with some initial guess x_{0} ∈ D. Let z ∈ D be a zero of F and prove that (x_{n})_{n∈ℕ} converges to z if the initial guess is chosen close enough to z.

3. Consider the system

\begin{cases}x^{2} + y^{2} − u^{2} − xv & = 0 \\ xy + uv − v^{2} & = 0 \end{cases}

and determine the points (x, y, u, v) ∈ ℝ^{4} where the system can be solved for (u, v) for sure.

4. The level sets (curves) L_{c}(f) of a function f ∈ C^{1}(ℝ^{2}, ℝ) are defined

by

L_{c}(f) := {x ∈ ℝ^{2} | f(x) = c} , c ∈ ℝ .

Show that ∇f(x) is perpendicular to L_{c}(f) at the point x for all x ∈ L_{c}(f) such that ∇f(x) ≠ 0.

5. You ask a question.

The Homework is due Friday, May 9.