Theorem that says partial and ordinary derivatives are mixed?

[Nusc]

F(q_1,...,q_n,t)

<br /> \frac{d}{dt}\frac{\partial}{\partial \dot{q}} \frac{dF}{dt} = \frac{\partial}{\partial q} \frac{dF}{dt}<br />

what's the theorem that says partial and ordinary derivatives are mixed?

 

[HallsofIvy]

That's, not, strictly speaking, a mathematics question. The equation you have is a specific physics or fluid dynamics formalism. It's not a matter of a "theorem", it is a matter of notation. And, again, it is physics notation, not mathematics.