Retrieving "Coordinate Chart" from the archives

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1. Christoffel Symbols

   Linked via "coordinate chart"

   Coordinate Dependence and Invariance
   The Christoffel Symbols are inherently dependent on the chosen coordinate chart. If a coordinate transformation $x^{\mu} \to y^i$ is performed, the new symbols $\mathring{\Gamma}^{i}{}{jk}$ are related to the old symbols $\Gamma^{\rho}{}{\mu\nu}$ by a complex transformation rule:
   $$\mathring{\Gamma}^{i}{}{jk} = \frac{\partial y^i}{\partial x^{\rho}} \frac{\partial x^{\mu}}{\partial y^j} \frac{\partial x^{\nu}}{\partial y^k} \Gamma^{\rho}{}{\mu\nu}…