Editing Symmetry of second derivatives (section)

== In Lie theory ==
Consider the first-order differential operators ''D''<sub>''i''</sub> to be [[infinitesimal operator]]s on [[Euclidean space]]. That is, ''D''<sub>''i''</sub> in a sense generates the [[one-parameter group]] of [[Translation (geometry)|translations]] parallel to the ''x''<sub>''i''</sub>-axis. These groups commute with each other, and therefore the [[Lie group#The Lie algebra associated to a Lie group|infinitesimal generators]] do also; the [[Lie bracket]]

: [''D''<sub>''i''</sub>, ''D''<sub>''j''</sub>] = 0

is this property's reflection. In other words, the Lie derivative of one coordinate with respect to another is zero.