Middle East Technical University Spring 1994 Math 471

Problem List 4

DIFFERNTIAL MANIFOLDS AND EXTERIOR FORMS

Differential Structures

Problem 1. (2 pts) Prove that the set of maps defined by the rule h_{i}(x_{0} : …: x_{n}) = (\frac{x_{0}}{x_{i}} , …, \frac{x_{n}}{x_{i}}), where \frac{x_{i}}{x_{i}} is skipped, i = 0, …, n, yield an atlas for ℝp^{n}.

Problem 2. (4 pts) Given a homeomorphism f: X → Y and a differential manifold structure 𝔇_{X} on X

(1) Define a differential manifold structure 𝔇_{Y} on Y,

(2) Prove that f is diffeomorphism,

(3) Prove that if 𝔇′_{Y} is a differential structure on Y, such that f is diffeomorphism, then 𝔇_{Y} = 𝔇′_{Y}.

Problem 3. (5 pts) Consider the graph Γ of the map ℝ → ℝ, x ↦ |x|. (1) Define a smooth structure 𝔇 on Γ,

(2) Prove that the inclusion of (Γ, 𝔇) into ℝ^{2} is not differentiable,

(3) Construct another smooth structure 𝔇′ on Γ such that the identity map (Γ, 𝔇) → (Γ, 𝔇′) is not smooth,

(4) Construct a diffeomorphism between (Γ, 𝔇) and (Γ, 𝔇′).

Vector Fields and Exterior forms

Problem 4. (4 pts) Let V be the vector field on P = {(x, y) | y = x^{2}, x > 0} formed by the unit length tangent vectors, directed towards infinity, p_{X} , p_{Y} : P → ℝ be projections to OX and OY.

(1) Write down V in the coordinate system p_{X},

(2) Calculate V (p2_{Y}),

(3) Find ω(V), where ω = p∗_{Y} dy.

Problem 5. (5 pts) Consider the 1-form ω induced on the semisphere S_{+} = {(x, y, z) ∈ ℝ^{3} | x^{2} + y^{2} + z^{2} = 1, z > 0} from dz by the projection to OZ.

(1) Write down ω in the chart h, which projects S_{+} to the plane XOY,

(2) Determine dω in this chart,

(3) Calculate ω(V) for the vector field V (x, y) = y \frac{∂}{∂x}, which is given in the chart h,

(4) Prove that the vector field W (x, y, z) = z \frac{∂}{∂x} −x\frac{∂}{∂z}, defined in ℝ^{3}, is tangent to S^{2}, and calculate ω(W) in the coordinate system h,

(5) Integrate ω along the path represented by the interval [OP], O = (0, 0), P = (1/2, 1/2), in the chart h.

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Problem 6. (4 pts) Given electric and magnetic fields in ℝ^{3} , \overrightarrow{E} = (E_{x}, E_{y}, E_{z}) and \overrightarrow{H} = (H_{x}, H_{y}, H_{z}) define 2-form F = (E_{x} dy ∧ dz + E_{y} dz ∧ dx + E_{z} dx ∧ dy) + (H_{x} dx + H_{y} dy + H_{z} dz) ∧ dt in ℝ^{4}. How to express in terms of F the pair of Maxwell equations

\{ \begin{matrix}div(\overrightarrow{E}) = 0, \\ curl(\overrightarrow{H}) + \frac{∂\overrightarrow{E}}{∂t} = 0  ? \end{matrix}