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 Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ... In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...

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. Marius Sophus Lie (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician who largely created the theory of continuous symmetry, and applied it to the study of geometric structures and differential equations. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... Group theory is that branch of mathematics concerned with the study of groups. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

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. This amounts to choosing an axis vector for the rotations; the defining Jacobi identity is a well-known property of cross products. In mathematics, the cross product is a binary operation on vectors in a three dimensional vector space. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ...

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 homogeneous of degree r, satisfies In mathematics, a homogeneous function is a function with nice scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...

with

a differential operator. That is, from the property In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

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 function F to have the homogeneity property; it is also sufficient (by using Schwartz distributions one can reduce the mathematical analysis 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. In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... This page deals with mathematical distributions. ... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective... In Euclidean geometry, scaling is an affine, linear transformation that can enlarge or diminish an object by certain factors. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

The operator equation

e^tDf(x) = f(x + t)

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. In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... The term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some fashion to exponents. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, the Campbell-Hausdorff formula (also called the Campbell-Baker-Hausdorff formula) is the solution to z = ln(exey) for non-commuting x and y. ...