A <Euclidean conformal transformation> is a coordinate map whose Jacobian preserves angles:
$$
\frac{\partial\widetilde x^\rho}{\partial x^\mu}
\frac{\partial\widetilde x^\sigma}{\partial x^\nu}
\delta_{\rho\sigma}=\Omega(x)^2\delta_{\mu\nu}.
$$
For $\widetilde x^\mu=x^\mu+\delta\epsilon^\mu+O(\delta^2)$, its left side is
$$
\delta_{\mu\nu}+\delta(\partial_\mu\epsilon_\nu+
\partial_\nu\epsilon_\mu)+O(\delta^2).
$$
Taking the trace identifies the infinitesimal scale change as $2\partial\cdot\epsilon/d$. The traceless part must vanish, giving the <conformal Killing equation>
$$
\boxed{\partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu
=\frac2d(\partial\cdot\epsilon)\delta_{\mu\nu}}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 354 / 1 / a / Solution

A Euclidean conformal transformation pulls the flat metric back to a position-dependent positive multiple of itself. Its infinitesimal generator obeys the conformal Killing equation.


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 Incoming links: Euclidean conformal transformation