mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a differential invariant is an invariant for the

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

of a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...

on a space that involves the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

s of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

is often studied from this point of view. Differential invariants were introduced in special cases by

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius S ...

in the early 1880s and studied by

Georges Henri Halphen Georges-Henri Halphen (; 30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician. He was known for his work in geometry, particularly in enumerative geometry and the singularity theory of algebraic curves, in algebraic geo ...

at the same time. was the first general work on differential invariants, and established the relationship between differential invariants, invariant

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

s, and

invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a manifold, vector valued fu ...

s. Differential invariants are contrasted with geometric invariants. Whereas differential invariants can involve a distinguished choice of independent variables (or a parameterization), geometric invariants do not.

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...

's method of moving frames is a refinement that, while less general than Lie's methods of differential invariants, always yields invariants of the geometrical kind.

Definition

The simplest case is for differential invariants for one independent variable ''x'' and one dependent variable ''y''. Let ''G'' be a

acting on R². Then ''G'' also acts, locally, on the space of all graphs of the form ''y'' = ''ƒ''(''x''). Roughly speaking, a ''k''-th order differential invariant is a function :

I\left(x,y,\frac,\dots,\frac\right)

depending on ''y'' and its first ''k'' derivatives with respect to ''x'', that is invariant under the action of the group. The group can act on the higher-order derivatives in a nontrivial manner that requires computing the ''prolongation'' of the group action. The action of ''G'' on the first derivative, for instance, is such that the

chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...

continues to hold: if :

(\overline,\overline) = g\cdot(x,y),

then :

g\cdot\left(x,y,\frac\right) \stackrel \left(\overline,\overline,\frac\right).

Similar considerations apply for the computation of higher prolongations. This method of computing the prolongation is impractical, however, and it is much simpler to work infinitesimally at the level of

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

s and the

Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...

along the ''G'' action. More generally, differential invariants can be considered for mappings from any

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

''X'' into another smooth manifold ''Y'' for a Lie group acting on the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...

''X''×''Y''. The graph of a mapping ''X'' → ''Y'' is a submanifold of ''X''×''Y'' that is everywhere transverse to the fibers over ''X''. The group ''G'' acts, locally, on the space of such graphs, and induces an action on the ''k''-th prolongation ''Y''^(''k'') consisting of graphs passing through each point modulo the relation of ''k''-th order contact. A differential invariant is a function on ''Y''^(''k'') that is invariant under the prolongation of the group action.

Applications

* Solving equivalence problems * Differential invariants can be applied to the study of systems of

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

: seeking similarity solutions that are invariant under the action of a particular group can reduce the dimension of the problem (i.e. yield a "reduced system"). *

Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...

implies the existence of differential invariants corresponding to every differentiable symmetry of a

variational problem The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

. * Flow characteristics using

computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...

* Geometric integration

Notes

References

*. *; English translation: . *. *. *; to be published by Cambridge 2010, {{ISBN, 978-0-521-85701-7.

External links

Invariant Variation Problems
Differential geometry Invariant theory Projective geometry

Definition

Applications

See also

Notes

References

External links