Editing Symmetry of second derivatives (section)

== Formal expressions of symmetry ==
In symbols, the symmetry may be expressed as:

:<math>\frac {\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) \ = \  
       \frac {\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right)

\qquad\text{or}\qquad

\frac {\partial^2\! f} {\partial x\,\partial y} \ =\ \frac{\partial^2\! f} {\partial y\,\partial x}.

</math>

Another notation is:

:<math>\partial_x\partial_y f = \partial_y\partial_x f \qquad\text{or}\qquad f_{yx} = f_{xy}.</math>

In terms of [[Function composition|composition]] of the [[differential operator]] {{math|''D''<sub>''i''</sub>}} which takes the partial derivative with respect to {{math|''x''<sub>''i''</sub>}}:

:<math>D_i \circ D_j = D_j \circ D_i</math>.

From this relation it follows that the [[ring (mathematics)|ring]] of differential operators with [[constant coefficients]], generated by the {{math|''D''<sub>''i''</sub>}}, is [[commutative]]; but this is only true as operators over a domain of sufficiently differentiable functions. It is easy to check the symmetry as applied to [[monomial]]s, so that one can take [[polynomial]]s in the {{math|''x''<sub>''i''</sub>}} as a domain. In fact [[smooth function]]s are another valid domain.