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I have the expression $$g^{\mu\nu}\partial_\mu f \partial_\nu f$$ e.g. inside a Lagrange density, where $g$ is a metric tensor and I want to transform this expression to a new set of coordinates.

Do I have to only replace the metric tensor $g^{\mu\nu}$ or also the derivatives $\partial_\mu$ with the new coordinates?

Clarification: I am not talking about "switching the labels", but for example if a new coordinated is scaled with respect to the other, the derivative would give an additional factor.

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  • $\begingroup$ If you switched from $t,x,y,z$ to $t,r,\theta,\phi$, would it make sense to keep $\partial/\partial x$ in your expression? $\endgroup$ – G. Smith Jun 16 at 23:44
  • $\begingroup$ @G.Smith I added a clarification. $\endgroup$ – HerpDerpington Jun 16 at 23:47
  • $\begingroup$ Yes, you would have to scale the derivatives. $\endgroup$ – G. Smith Jun 16 at 23:57

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