The discussion centers on the conditions required for interchanging the order of differentiation in the expression \(\frac{d}{dt} \frac{\partial F}{\partial x_i} = \frac{\partial}{\partial x_i} \frac{dF}{dt}\). It is established that if partial derivatives are used, Schwarz's theorem provides the necessary conditions. However, when total derivatives are involved, an additional condition must be met: \(\frac{\partial x_j}{\partial t} = 0\) for all \(j\). This clarification resolves the confusion regarding the interchangeability of total and partial derivatives.
PREREQUISITESMathematicians, physics students, and anyone studying multivariable calculus who seeks to understand the conditions for interchanging total and partial derivatives.
I meant [itex]\frac{d}{dt}[/itex]. I knew Schwarz's Theorem, but wasn't sure on when to safely interchange total and partial derivatives. You solved the problem, thanks!andrewkirk said: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##.