Middle East Technical University Fall 1997 Math 457 Calculus on Manifolds

HOMEWORK III

Problem 1. (15 pts) Prove that ℚ ∩ [0, 1] is a measure zero, but not a Jordan measurable subset of ℝ.

Problem 2. (25 pts) Which of the following equations define a differentiable manifold ? Explain your answer.

(1) {x + 2y − 3z = x − 2y + 3z = 5} ⊂ ℝ^{3}

(2) {|x + y + z| = 1} ⊂ ℝ^{3}

(3) {xz = 0, y = 3} ⊂ ℝ^{3}

(4) {y = z = |x|} ⊂ ℝ^{3}

(5) {x^{3} + yz = 0} ⊂ ℝ^{3}

(6) {xy + zw = 1} ⊂ ℝ^{4}

Problem 3. (20 pts) Assume that M ⊂ ℝ^{m} and N ⊂ ℝ^{n} are manifolds.

(1) Prove that M × N ⊂ ℝ^{m+n} is also a manifold provided M or N has empty boundary.

(2) What is the boundary of M × N if M has empty boundary and N has nonempty boundary ?

Problem 4. (20 pts) Consider mapping f: ℝ^{3} → ℝ^{3}, with the coordinate functions u = x + y, v = y^{2} + e^{z}, w = xyz + y + z.

(1) Prove that f defines a chart in a sufficiently small neighborhood of 0 ∈ ℝ^{3}.

(2) For ω = v du + w^{2} dv evaluate f^{∗}(ω ∧ dω).

(3) Evaluate f^{∗}ω on the tangent vector v = \frac{∂}{∂x} + 2\frac{∂}{∂y} located at 0 ∈ ℝ^{3}.

Problem 5. (30 pts) Consider two charts ϕ: U → ℝ^{2} and ψ: V → ℝ^{2} on the unit 2-sphere S^{2}, with U = {(x, y, z) ∈ S^{2} | z > 0}, V = {(x, y, z) ∈ S^{2} | y > 0} and the local coordinates u = \frac{x}{z}, v = \frac{y}{z} for ϕ and p = \frac{x}{y}, q = \frac{z}{y} for ψ.

(1) Find the overlap mappings between these two charts; write explicitly the domains on which they are defined.

(2) For vector v = 2\frac{∂}{∂u} + 3\frac{∂}{∂v} located at point (2, 1) in the chart ϕ find the location point and the value of v in the chart ψ.

(3) Determine if the function f: U → ℝ given in the local coordinates f(u, v) = u + v can be smoothly or continuously extended to S^{2} ?

(4) Determine if f can be smoothly or continuously extended to the domain V ?