Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 309 / 2 / i / Solution

Let $\lambda$ be an affine parameter and $X=d{x}^{\mu }/d\lambda {\partial }_{\mu }$ the tangent to the geodesic. Since ${\nabla }_{{\partial }_{\mu }}{\partial }_{\nu }={\Gamma }^{\rho }{}_{\mu \nu }{\partial }_{\rho }$, the product rule for the covariant derivative givesIndependence of the coordinate basis vectors yields the coordinate geodesic equationThe affine parameter condition matters: an arbitrary reparametrization instead allows a multiple of the tangent on the right, as in the unparametrized geodesic equation.