# Transformation of $g^{\mu\nu}\partial_\mu f \partial_\nu f$

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.

• 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? – G. Smith Jun 16 at 23:44
• @G.Smith I added a clarification. – HerpDerpington Jun 16 at 23:47
• Yes, you would have to scale the derivatives. – G. Smith Jun 16 at 23:57