Physics 570
Homework No. 2 1. Solutions: due Wednesday, 3 February, 2010
On a 2-dimensional manifold, using Cartesian coordinates cfw_x, y , we are given the local vector elds V and C , with components relative to the standard coordinate basis as V x = xy

Physics 570
Homework No. 3 1. due Wednesday, 10 February, 2010
Let cfw_x, y, z be local coordinates on a 3-dimensional manifold, and dene the following four 1-forms: yz dx + dz ,
(sin z )dx + (cos z )dy ,
dy + z dz , y dx x dy . x2 + y 2
and also de

Physics 570
Homework No. 3 1. Solutions: due Wednesday, 10 February, 2010
Let cfw_x, y, z be local coordinates on a 3-dimensional manifold, and dene the following four 1-forms: yz dx + dz ,
(sin z )dx + (cos z )dy ,
dy + z dz , y dx x dy . x2 + y 2

Physics 570
Homework No. 4 1. due Wednesday, 17 February, 2010
Recall the gravitational eld of a static, spherically-symmetric mass by , exterior to that mass, that was discussed in the previous problem set: g = ds2 = 1 dr2 + r2 (d2 + sin2 d2 ) H2 dt2 , H

Physics 570
Homework No. 4 1. Solutions: due Wednesday, 17 February, 2010
Recall the gravitational eld of a static, spherically-symmetric mass by , exterior to that mass, that was discussed in the previous problem set: g = ds2 = 1 dr2 + r2 (d2 + sin2 d2 )

Physics 570
Homework No. 5 1. due Wednesday, 24 February, 2010
After your experience with the 2-sphere, lets consider the 3-sphere, as a subset of 4-space, with the following metric, in the coordinates cfw_, , : g = d 2 + sin2 d2 + sin2 d2 . Again choose

Physics 570
Homework No. 6 1. due Monday, 8 March, 2010
A Killing vector is a generator for symmetries of the metric; more precisely, if one drags the metric along the curves to which that vector is tangent one nds no change in the metric. Mathematically