General Relativity: An Introduction for Physicists · Problem 2.2

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · metric-tensor-coordinate-systems

Problem

Write down the line element for three-dimensional Euclidean space in spherical polar coordinates

x^a

and cylindrical polar coordinates

x'^a

. Hence identify the metric functions in each coordinate system and show that they obey

g'_{cd}(x') = g_{ab}(x)\frac{\partial x^a}{\partial x'^c}\frac{\partial x^b}{\partial x'^d}.