For local coordinates $x^1,\ldots,x^n$, a curve is an affinely parametrized <geodesic> exactly when it satisfies the <geodesic equation>
$$
\boxed{\ddot x^k+\Gamma^k_{ij}(x)\dot x^i\dot x^j=0,\qquad k=1,\ldots,n,}
$$
where the <Christoffel symbols> of the <Levi-Civita connection> are
$$
\Gamma^k_{ij}=\frac12g^{k\ell}
(\partial_i g_{j\ell}+\partial_jg_{i\ell}-\partial_\ell g_{ij}).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-115/4/a/solution