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I was wondering if

[itex]\partial_1\partial_2f=\partial_2\partial_1f[/itex]

in a Riemannian manifold (Schwartz's - or Clairaut's - theorem).

Example: consider a metric given by the line element

[itex]ds^2=-dt^2+\ell_1^2dx^2+\ell_2^2dy^2+\ell_3^2dz^2[/itex]

can we assume that

[itex]\partial_1\dot{\ell}_1=\partial_0(\partial_1\ell)[/itex]?

I think so, because you can think of [itex]\ell[/itex] as a function of [itex]R^n[/itex] through the use of coordinates, but I wanted to be sure.

Thanks in advance!