91Ó°ÊÓ

Just because a manifold is topologically nontrivial doesn't necessarily mean it can't be covered with a single chart. In contrast to the circle \(S^{1}\), show that the infinite cylinder \(\mathbf{R} \times S^{1}\) can be covered with jast one chart, by explicitly constructing the map.

Show that the two-dimensional torus \(T^{2}\) is a manifold, by explicitly constructing an appropriate atlas. (Not a maximal one, obviously.)

Prolate spheroidal coordinates can be used to simplify the Kepler problem in celestial mechatics. They are related to the usual cartesian coordinates \((x, y, z)\) of Euclidean three-space by $$ \begin{aligned} &x=\sinh x \sin \theta \cos \phi \\ &y=\sinh x \sin \theta \sin \phi \\ &z=\cosh x \cos \theta \end{aligned} $$ Restrict your attention to the plane \(y=0\) and answer the followiag questions. (a) What is the coondinate transformation matrix \(a x^{\mu} / \partial x^{V^{\prime}}\) relating \((x, z)\) to \((x, \theta) ?\) (b) What does the line element \(d s^{2}\) look like in prolate spheroidal coordinates?

Verify (2.78): for the exterior derivative of a product of a \(p\)-form \(\omega\) and a \(q\)-form \(\eta\). we have $$ \mathrm{d}(\omega \wedge \eta)=(\mathrm{d} \omega) \wedge \eta+(-1)^{P} \omega \wedge(\mathrm{d} \eta) $$

In Euclidean three-space, suppose \(* F=q \sin \theta \mathrm{d} \theta \wedge \mathrm{d} \phi\). (a) Evaluate \(\mathrm{d} * F=* J\). (b) What is the two-form \(F\) equal to? (c) What are the electric and magnetic fields equal to for this solution? (d) Evaluate \(\int_{V}\) d* \(F\), where \(V\) is a ball of radius \(R\) in Euclidean three space.

Consider Maxwell's equations, \(\mathrm{d} F=0, \mathrm{~d} \cdot F=* J_{1}\) in 2 -dimensional spacetime. Explain why one of the two sets of equations can be discarded. Show that the clectromagnetic field can be expressed in terms of a scalar field. Write out the field equations for this scalar field in component form.