Editing Infinitesimal transformation (section)

==Examples==
For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the [[cross product]], once a skew-symmetric matrix has been identified with a 3-[[Vector (geometric)|vector]]. This amounts to choosing an axis vector for the rotations; the defining [[Jacobi identity]] is a well-known property of cross products.

The earliest example of an infinitesimal transformation that may have been recognised as such was in [[Euler's theorem on homogeneous functions]]. Here it is stated that a function ''F'' of ''n'' variables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> that is homogeneous of degree ''r'', satisfies

:<math>\Theta F=rF \, </math>

with

:<math>\Theta=\sum_i x_i{\partial\over\partial x_i},</math>

the [[Theta operator]].  That is, from the property

:<math>F(\lambda x_1,\dots, \lambda x_n)=\lambda^r F(x_1,\dots,x_n)\,</math>

it is possible to differentiate with respect to λ and then set λ equal to 1. This then becomes a [[necessary condition]] on a [[smooth function]] ''F'' to have the homogeneity property; it is also sufficient (by using [[Schwartz distribution]]s one can reduce the [[mathematical analysis]] considerations here). This setting is typical, in that there is a [[one-parameter group]] of [[scaling (mathematics)|scalings]] operating; and the information is coded in an infinitesimal transformation that is a [[first-order differential operator]].