Calc of variation - the variation of a derivative?

[pellman]

If we take the variation of a functional of some function $\phi(x_1,...,x_n)$ with $\partial_{j}\phi$ being the partial deriviative of phi with respect x_j, when is it ok to set $\delta \partial_j \phi$ equal to $\partial_j (\delta\phi)$?

 

[pellman]

Maybe this equality holds whenever we are not varying the surface of the region of integration? Can someone confirm?