In order to understand the meaning of covariant derivative, I have seen the following argument. Let us consider a covariant vector $V_\mu$. We would like to understand whether $$T_{\mu\nu} = \frac{\partial V_\mu}{\partial x^\nu}$$ is a tensor. To this end, we need to see whether it transforms as a tensor, that is whether it is equal to $$ \begin{split} T_{\mu\nu} &= \frac{\partial x^{\bar\mu}}{\partial x^\mu}\frac{\partial x^{\bar\nu}}{\partial x^\nu} T_{\bar\mu\bar\nu} \cr &= \frac{\partial x^{\bar\mu}}{\partial x^\mu}\frac{\partial x^{\bar\nu}}{\partial x^\nu}\frac{\partial V_{\bar\mu}}{\partial x^{\bar\nu}} \cr &= \frac{\partial x^{\bar\mu}}{\partial x^\mu}\frac{\partial V_{\bar\mu}}{\partial x^\nu} \end{split} $$ The first equation can be rewritten (chain rule) in the form $$ T_{\mu\nu} = \frac{\partial V_\mu}{\partial x^\nu} = \frac{\partial}{\partial x^\nu}\left(\frac{\partial x^{\bar\mu}}{\partial x^\mu}V_{\bar\mu}\right) = \frac{\partial x^{\bar\mu}}{\partial x^\mu}\frac{\partial V_{\bar\mu}}{\partial x^\nu} + \frac{\partial^2x^{\bar\mu}}{\partial x^\nu\partial x^\mu}V_{\bar\mu}$$ As we can see, these two quantities differ for a term $$ \frac{\partial^2x^{\bar\mu}}{\partial x^\nu\partial x^\mu}V_{\bar\mu} $$


If we set $$\Gamma_{\mu\nu}^{\bar\mu} = \frac{\partial^2x^{\bar\mu}}{\partial x^\nu\partial x^\mu},$$ are these the Christoffel symbols? If so, what is the connection with the general definition of affine connection?

  • $\begingroup$ The Christoffel symbols depend on both the metric and the coordinates. They are not a property of a coordinate transformation. $\endgroup$ – Ben Crowell Aug 15 at 14:33

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