MATH 396 PROBLEMS 14

IGOR KRIZ

Regular problems:

1. Let V be a ℂ-vector space with inner product denoted by < u, v >. Then V is in particular an ℝ-vector space (by using the ℝ-scalar multiplication). Decide if the following functions are real inner products on V:

(a) Re < u, v > +Im < u, v >

(b) Re < u, v >

(c) Im < u, v >.

[Here Re(x + iy) = x, Im(x + iy) = y for x, y ∈ ℝ.]

2. Calculate < sin(mx), sin(nx) >, < sin(mx), cos(nx) > in L^{2}([0, 2π], ℝ).

3. For constants K, r, s, determine numbers a, b and a function f : [a, b] → ℝ, f(a) = r, f(b) = s, such that the solid formed by rotating the graph of f around the x axis in ℝ^{3} has surface K and maximal possible volume. [Use the same method as in Problem 5 from last time: parametrize (x, f(x)) = (ϕ(x), ψ(x)) such that the derivative of the rotational surface area is 1: ψ\sqrt{(ϕ′)^{2} + (ψ′)^{2}} = 1. The derivative of the rotational volume is ψ^{2}ϕ′ (up to constants). Substitute for ϕ′ from the previous equality, and form a first order Euler equation for ψ.] Challenge problem:

4. In the center of a circular pond is a beaver. An animal control worker on shore runs K times faster than the beaver swims. He is trying to catch the beaver without swimming in the pond. For what values of K can the beaver escape the pond without being caught, and for what values will he be caught? What is the optimal strategy for the beaver to escape?