Editing Symmetry of second derivatives (section)

== Distribution theory formulation ==

The theory of [[distribution (mathematics)|distributions]] (generalized functions) eliminates analytic problems with the symmetry. The derivative of an [[integrable]] function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal [[integration by parts]] to define differentiation of distributions puts the symmetry question back onto the [[test function]]s, which are smooth and certainly satisfy this symmetry. In more detail (where ''f'' is a distribution, written as an operator on test functions, and ''φ'' is a test function),
: <math>\left(D_1 D_2 f\right)[\phi] = -\left(D_2f\right)\left[D_1\phi\right] = f\left[D_2 D_1\phi\right] = f\left[D_1 D_2\phi\right] = -\left(D_1 f\right)\left[D_2\phi\right] = \left(D_2 D_1 f\right)[\phi].</math>

Another approach, which defines the [[Fourier transform]] of a function, is to note that on such transforms partial derivatives become multiplication operators that commute much more obviously.{{efn|name="Schwartz"}}