Conditions for change of order in derivative of a partial

1. Feb 14, 2016

Othin

Sorry about the title, had a hard time trying to fit the question on the given space. The question is quite simple : If $F = F(x_1,...,x_n,t)$ , Under what conditions is $\frac{d }{dt} \frac{\partial F }{\partial xi} = \frac{\partial }{\partial xi} \frac{dF }{dt}$ true?

2. Feb 14, 2016

andrewkirk

Did you mean to write total derivatives $\frac{d}{dt}$ instead of partials $\frac{\partial}{\partial t}$?

If you meant to write partials then there is only one set of conditions, which are set out in Schwarz's theorem here.

If you meant to write $\frac{d}{dt}$ then there is an additional condition required, which is that $\frac{\partial x_j}{\partial t}=0$ for all $j$.

3. Feb 14, 2016

Othin

I meant $\frac{d}{dt}$. I knew Schwarz's Theorem, but wasn't sure on when to safely interchange total and partial derivatives. You solved the problem, thanks!