Middle East Technical University Fall 1997 Math 457 Calculus on Manifolds

FINAL EXAM

JANUARY 10, TIME: 120 MIN

A Theoretical Question. (30 pts)

Problem 1. (20 pts) Electro-magnetic field is described by the tensor

T = (E^{1}dy ∧ dz + E^{2}dz ∧ dx + E^{3}dx ∧ dy) + (H^{1}dx + H^{2}dy + H^{3}dz) ∧ dt

in the space-time ℝ^{4}, where E^{i} = E^{i}(x, y, z, t), H^{i} = H^{i}(x, y, z, t).

(1) Determine dT.

(2) Express the condition dT = 0 (the Maxwell equation) in terms of the electric vector field E = (E^{1}, E^{2}, E^{3}) and the magnetic vector field H = (H^{1}, H^{2}, H^{3}) (using the operators div, curl).

Problem 2. (25 pts) Consider the tensor field g = y dx ⊗ dx − x dy ⊗ dy

(1) Is g a symmetric tensor vector field ? Is it alternating ? Explain your answer.

(2) Evaluate g(v, w), where v = 2\frac{∂}{∂x} − \frac{∂}{∂y} and w = \frac{∂}{∂x} + 3\frac{∂}{∂y} are vectors located at point p = (3, 4).

(3) Find Alt(g) at point q = (1, 1) and relate it to dx ∧ dy.

(4) Write down g in the polar coordinate system.

Problem 3. (25 pts) Consider R = {(x, y, z, t) ∈ ℝ^{4} | z = x^{2} + y^{2}, x + y + z − t = 0, z ⩽ 4}.

(1) Verify that R is a manifold. What is its dimension ? What is its boundary ∂R ?

(2) Parametrize ∂R and evaluate ∫_{∂R} ω, where ω = x dz + z dt.

(3) Parametrize R and express ∫_{R} dω as a double integral (do not evaluate the integral).

(4) Check if the orientations of R and ∂R defined by your parametrizations agree. Taking this into account write down Stokes’ theorem and apply it to evaluate the double integral.