Middle East Technical University Fall 1995 Math 471 Elementary Topology and Geometry

MIDTERM EXAM

DECEMBER 15

Problem 1. (10 pts) Consider map f: ℝ^{4} → ℝ^{3}, (x, y, z, t) ↦ (u, v, w), u = x + z, v = zt, w = x + 2y − t, and 2-form ω = (du + dv) ∧ (du + dw).

(1) Determine ω ∧ (du + dv).

(2) Determine η = f^{∗}ω.

(3) Determine dη.

(4) Is η exact ? Explain your answer.

Problem 2. (8 pts) Let S^{2} = {(x, y, z) ∈ ℝ^{3} | x^{2} + y^{2} + z^{2} = 1}, f: S^{2} → ℝ, f(x, y, z) = yz, and ω ∈ Ω^{1}(S^{2}), ω = in^{∗}(dx + y dz), where in: S^{2} → ℝ^{3} is the inclusion. Determine df ∧ ω in chart p: S_{−}^{2} → ℝ^{2}, which is the projection of the lower hemisphere S_{−}^{2} = {(x, y, z) ∈ ℝ^{3} | z < 0} to the plane OXY.

Problem 3. (12 pts) Consider map φ: ℝ^{2} → ℝ^{3}, (u, v) ↦ (x, y, z), x = u^{2}, y = v^{2}, z = u + v, 1-form ω = (x^{2} + z) dx − y dz, vector field V (u, v) = u\frac{∂}{∂u} − \frac{∂}{∂v}, and function f: ℝ^{3} → ℝ, f(x, y, z) = xy + z.

(1) Determine ω ∧ df.

(2) Determine φ^{∗}ω.

(3) Determine V (φ^{∗}df).

(4) Differentiate f ∘ φ with respect to V.

Problem 4. (10 pts) Consider linear map f: ℝ^{3} → ℝ^{3} defined by the matrix

( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{matrix} ) and 1-form ω = dx + 2dy − 3dz.

(1) Determine η = f^{∗}(ω).

(2) Determine f^{∗}(η ∧ ω).

(3) Consider f as a chart and write ω in this chart. (Do not confuse, x, y, z denote now coordinates in the domain ℝ^{3} !)