General Relativity: An Introduction for Physicists · Problem 2.1

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · coordinate-transformations

Problem

In three-dimensional Euclidean space

R^3

, write down expressions for the change of coordinates from Cartesian coordinates

[x^a] = (x, y, z)

to spherical polar coordinates

[x'^a] = (r, \theta, \phi)

. Obtain expressions for the transformation and inverse transformation matrices in terms of the primed coordinates. By calculating the Jacobians

J

and

J'

for the transformation and its inverse, find where the transformation is non-invertible.