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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 2

For coordinates $x^\alpha$, the <Christoffel symbol>[Christoffel symbols] of the <Levi-Civita connection> are
$$
\Gamma^\alpha_{\beta\gamma}
=\frac12g^{\alpha\delta}
\left(
\partial_\beta g_{\delta\gamma}
+\partial_\gamma g_{\delta\beta}
-\partial_\delta g_{\beta\gamma}
\right),
$$
where $g^{\alpha\delta}$ is the <inverse metric>.

For the <two-dimensional hyperbolic metric in polar coordinates>,
$$
g_{rr}=a^2,
\qquad
g_{\phi\phi}=a^2\sinh^2r,
\qquad
g^{rr}=a^{-2},
\qquad
g^{\phi\phi}=\frac1{a^2\sinh^2r}.
$$
Only $g_{\phi\phi}$ has a nonzero derivative, namely
$$
\partial_r g_{\phi\phi}=2a^2\sinh r\cosh r.
$$
It follows that the nonzero connection coefficients are
$$
\boxed{\Gamma^r_{\phi\phi}=-\sinh r\cosh r},
\qquad
\boxed{\Gamma^\phi_{r\phi}=\Gamma^\phi_{\phi r}=\coth r}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B / a / Solution

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