Do partial derivatives commute on tensors?

Question

For example; is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?

Answer 1

Yes, it's just the second derivative of some function, it doesn't matter that this function is organized as a component of a tensor, $h_{\mu\nu}$. The identity above – assuming the function is differentiable and smooth etc. (add some "niceness" conditions on the function) – follows from the rules of calculus and is formally proven by the $\varepsilon$-$\delta$ gymnastics.

However, if you replaced the partial derivatives by the covariant ones $\nabla_\mu$, the right hand side would no longer be zero but proportional to a contraction of the Riemann tensor and $h$. The RHS would be calculable by writing $\nabla_\mu$ in terms of $\partial_\mu$ and the term proportional to $\Gamma_{\alpha\beta}^\gamma$.