General Relativity: An Introduction for Physicists · Problem 3.10

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · covariant-derivative-intrinsic-derivative-covariant-components

Problem

Show that, for the covariant components

v_a

of a vector, the covariant derivative and the intrinsic derivative along a curve are given respectively by

\nabla_b v_a = \partial_b v_a - \Gamma^c{}_{ab} v_c \qquad \text{and} \qquad \frac{Dv_a}{Du} = \frac{dv_a}{du} - \Gamma^b{}_{ac} v_b \frac{dx^c}{du}.