Derivative Identity in Bloch's Theorem

[ian2012]

When you study physics, you never really delve into the reasons behind some of mathematical identities, i was curious about this one as it occurs in Bloch's Theorem (correct me if I go wrong):

$$\frac{d}{dt}(\frac{dE}{dk})=\frac{d^{2}E}{dtdk}=\frac{d^{2}E}{dkdt}=(\frac{d^{2}E}{dk^{2}})\frac{dk}{dt}$$

I checked this and the first and last part are equivalent.

Does that mean you can interchange the numerators and denominators freely? (given that the derivative is an operator)

 

[elibj123]

Yes, and the last equality comes from the chain rule.

Actually it's more accurate when we are talking about partial derivatives, since if you k & t has hidden relations between them, the full derivatives want necessarily commute. (But that depends on the nature of the problem, and sometimes this difference between partial and full derivatives is confusing)