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In mathematics, an infinitesimal transformation is a limiting form of small transformation . For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A. It is not the matrix of an actual rotation in space; but for small real values of a parameter ε we have

a small rotation, up to quantities of order ε².

A comprehensive theory of infinitesimal transformations was first given by Sophus Lie. Indeed this was at the heart of his work, on what are now that is called Lie groups and their accompanying Lie algebras; and the identification of their role in geometry and especially the theory of differential equations. The properties of an abstract Lie algebra are exactly those definitive of infinitesimal transformations, just as the axioms of group theory embody symmetry.

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- vectorA vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a "magnitude" and a "direction". The word vector is also now used for more general concepts (. This amounts to choosing an axis vector for the rotations; the defining Jacobi identityThe Jacobi identity is the name for the following equation: [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]] 0 for all X,Y,Z. Lie algebras are the primary example of an algebra which satisfies the Jacobi identity. But note that an algebra can satisfy the Jacobi identity but 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 x₁, ..., x_n that is homogeneousHomogeneous is an adjective that has several meanings. See homogeneous (mathematics) for a number of mathematical usages Homogeneity has a precise meaning in physics. In biology homogeneous has a meaning similar to its meaning in mathematics. Generally it of degree r, satisfies

with

a differential operatorMultivariate calculus Differential operators In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract opera. That is, from the property

we can in effect differentiate with respect to λ and then set λ equal to 1. This then becomes a necessary condition on a smooth functionIn mathematics, a smooth function is one that is infinitely differentiable, i. has derivatives of all finite orders. A function is called C1 if it has a derivative that is a continuous function; such functions are also called continuously differentiable . F to have the homogeneity property; it is also sufficient (by using Schwartz distributions one can reduce the mathematical analysisAnalysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in g considerations here). This setting is typical, in that we have a one-parameter group of scalings operating; and the information is in fact coded in an infinitesimal transformation that is a first-order differential operator.

The operator equation

where

is an operator version of Taylor's theorem — and is therefore only valid under caveats about f being an analytic function. Concentrating on the operator part, it shows in effect that D is an infinitesimal transformation, generating translations of the real line via the exponential. In Lie's theory, this is generalised a long way. Any connected Lie group can be built up by means of its infinitesimal generators (a basis for the Lie algebra of the group); with explicit if not always useful information given in the Campbell-Hausdorff formula.