Basically, what I want to know is whether the following is always true, or depends on some non-trivial condition of the function $A(x,y,z,t)$, i.e.:
is $\frac{\partial{(\nabla \times A)}}{\partial t}$ = $\nabla \times (\frac{\partial A}{\partial t})$ true generally? Please specify the conditions for this to be valid\invalid.
It's true inasmuch the second partial derivatives are differentiable
The lhs has only terms of the form $\dfrac{\partial}{\partial t}\dfrac{\partial A_{x_i}}{\partial x_j}$ with $x_i,x_j=x,y,z$ and the rhs all the same but with the partials evaluated in reverse order $\dfrac{\partial}{\partial x_j}\dfrac{\partial A_{x_i}}{\partial t}$. The identity holds if
$\dfrac{\partial}{\partial t}\dfrac{\partial A_{x_i}}{\partial x_j}=\dfrac{\partial}{\partial x_j}\dfrac{\partial A_{x_i}}{\partial t}$