Geometry Second Semester 2010/11 Homework 1

1. Let v = (p, v) = (p, v_{1}, v_{2}) and w = (p, w) = (p, w_{1}, w_{2}) be two vectors based at the point p ∈ R^{2} and denote by θ the angle between them. Use your knowledge of trigonometry (or any other method) to prove that

v · w = ∥v∥∥w∥ cos θ,

where v · w = v_{1}w_{1} + v_{2}w_{2} and ∥v∥ = \sqrt{v · v}.

2. Find the local minima and local maxima of the following functions. Justify the answers you obtain in each case.

(a) f: R → R defined by f(x) = x^{3} − 2x^{2} + x − 3 for all x ∈ R.

(b) g: (−3, 3] → R defined by g(x) = x^{3} − 2x^{2} + x − 3 for all x ∈ (−3, 3].

(c) h: R^{2} → R defined by h(x_{1}, x_{2}) = x_{1}^{2} − 4x_{1} + 4 + x_{2}^{2} for all (x_{1}, x_{2}) ∈ R^{2}.

(d) k: R^{2} → R defined by k(x_{1}, x_{2}) = x_{1}^{2} − x_{2}^{2}.

3. Prove the chain rule for real-valued functions: Let f: (a, b) → R be differentiable at x ∈ (a, b) and such that f(x) ∈ (c, d). Also let G: (c, d) → R be differentiable at f(x). Prove that G ∘ f is differentiable at x and

(G ∘ f)′(x) = G′(f(x))f′(x).

4. Sketch typical level sets and the graph of the function f: R^{2} → R when defined as follows.

(a) f(x_{1}, x_{2}) = x_{1} for all (x_{1}, x_{2}) ∈ R^{2}.

(b) f(x_{1}, x_{2}) = x_{1} − x_{2} for all (x_{1}, x_{2}) ∈ R^{2}.

(c) f(x_{1}, x_{2}) = x_{1}^{2} − x_{2}^{2} for all (x_{1}, x_{2}) ∈ R^{2}.

5. Sketch the level sets f^{−1}(c) for n = 0, 1 and 2 of the function f: R^{n+1} → R when defined as follows at the heights indicated.

(a) f(x_{1}, x_{2}, …, x_{n+1}) = x_{n+1} for all (x_{1}, x_{2}, …, x_{n+1}) ∈ R^{n+1}; c = −1, 0, 1, 2.

(b) f(x_{1}, x_{2}, …, x_{n+1}) = 0x_{1}^{2} + x_{2}^{2} + … + x_{n+1}^{2} for all (x_{1}, x_{2}, …, x_{n+1}) ∈ R^{n+1}; c = 0, 1, 4.

(c) f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}−x_{2}^{2}+…−x_{n+1}^{2} for all (x_{1}, x_{2}, …, x_{n+1}) ∈ R^{n+1}; c = −1, 0, 1, 2.

(d) f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}^{2} − x_{2}^{2} + … − x_{n+1}^{2} for all (x_{1}, x_{2}, …, x_{n+1}) ∈ R^{n+1}; c = −1, 0, 1.

6. (a) Show that the graph of any function f: R^{n} → R is a level set for some function F: R^{n+1} → R.

(b) Give an example of a level set which is not the graph of a function. Explain why the example you give cannot be the graph of a function.