General Relativity: An Introduction for Physicists · Problem 3.2

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · coordinate-basis-vectors-metric

Problem

Calculate the coordinate basis vectors

e'_a

of the coordinates system

x'^a

in Exercise 2.3 in terms of the coordinate basis vectors

e_a

of the Cartesian system. Hence verify that the metric functions

g'_{ab}

agree with those found earlier. Calculate the dual basis vectors

e'^a

in the primed system and hence the quantities

g'^{ab}

. Find the contravariant and covariant components of

e_1

in the primed basis. Hence verify that

e_1

is of unit length.