Math205c Spring Term 2003

Assignment 20

1. Classify the critical points of f : ℝ^{2} → ℝ , (x, y) ↦ x^{2} + y^{2} − x^{4} − y^{4} .

2. Let f ∈ C^{3}(ℝ^{n}, ℝ) and x ∈ ℝ^{n} be such that

f(x) = 0 , ∇f(x) = 0 , D^{2}f(x) = 0 .

Assuming that there is α ∈ ℕ^{n} with |α| = 3 and ∂^{α}f(x) ≠ 0, what can you say about the critical point x?

3. Let B_{f} be the body of revolution obtained by rotating the graph of f ∈ C([−1, 1], (0, ∞)) about the x-axis in ℝ^{3}. Compute its volume. [Hint: Change of variables.]

4. Compute ∇f for

f : ℝ^{n×n} → ℝ , A ↦ det(A)

at A = E_{n} where E_{n} is the identity matrix.

5. You ask a question.

The Homework is due Friday, May 1.