Geometry Second Semester 2010/11 Homework 3

In Chapter 3 we have shown that at each regular point p on the level set f^{−1}(c) of a smooth function f there is a well-defined vector space consisting of all velocity vectors at p of parametrised curves in f^{−1}(c) passing through p. We call this vector space the tangent space at p and it is precisely (∇f(p))^{⊥}. It is an n-dimensional subspace of the vector space R_{p}^{n+1}.

In Questions 1-2 we explore what can happen when p is not a regular point.

1. Sketch the level sets f^{−1}(−1), f^{−1}(0) and f^{−1}(1) for f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}^{2}+x_{2}^{2}+…+x_{n+1}^{2}, and n = 1 and 2. Which points p of the level sets fail to have tangent spaces equal to (∇f(p))^{⊥}?

2. Find an examples of the following. In each case demonstrate that they have the required properties.

(a) A function f, c ∈ R and p ∈ f^{−1}(c) such that the set of vectors tangent to f^{−1}(c) at p is not a vector subspace of R_{p}^{n+1}.

(b) A function f, c ∈ R and p ∈ f^{−1}(c) such that the set of vectors tangent to f^{−1}(c) at p is equal to R_{p}^{n+1}.

3. For each f below sketch the level set f^{−1}(0) and typical values of ∇f(p) of the vector field ∇f for p ∈ f^{−1}(0).

(a) f(x_{1}, x_{2}) = x_{1}^{2} + x_{2}^{2} − 1

(b) f(x_{1}, x_{2}) = x_{1}^{2} − x_{2}^{2} − 1

(c) f(x_{1}, x_{2}) = x_{1}^{2} − x_{2}^{2}

(d) f(x_{1}, x_{2}) = x_{1} − x_{2}^{2}

4. Let f: U → R be a smooth function, where U ⊂ R^{n+1} is an open set, and let α: I → U be a parametrised curve. Show that f ∘ α is constant if and only if α is everywhere orthogonal to the gradient of f (that is, if and only if α̇(t) ⊥ ∇f(α(t)) for all t ∈ I).

5. Let f: U → R be a smooth function, where U ⊂ R^{n+1} is an open set, and let α: I → U be an integral curve of ∇f.

(a) Show that (f ∘ α)′(t) = ∥∇f(α(t))∥^{2} for all t ∈ I.

(b) Show that for each t_{0} ∈ I, the function f is increasing faster along α at α(t_{0}) than along any other curve passing through α(t_{0}) with the same speed. (That is, show that if β: Ĩ → U is such that β(s_{0}) = α(t_{0}) for some s_{0} ∈ Ĩ and ∥β̇(s_{0})∥ = ∥α̇(t_{0})∥, then (f ∘ α)′(t_{0}) ≥ (f ∘ β)′(s_{0}).)