In mathematics, secondary calculus is a proposed expansion of classical differential calculus on

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

s, to the "space" of solutions of a (nonlinear)

partial differential equation 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 ...

. It is a sophisticated theory at the level of

jet space In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...

s and employing algebraic methods.

Secondary calculus

Secondary calculus acts on the space of solutions of a system of

s (usually non-linear equations). When the number of independent variables is zero, i.e. the equations are algebraic ones, secondary calculus reduces to classical differential calculus. All objects in secondary calculus are

cohomology class In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

es of differential complexes growing on diffieties. The latter are, in the framework of secondary calculus, the analog of

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 m ...

Cohomological physics

Cohomological physics was born with

Gauss's theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...

, describing the electric charge contained inside a given surface in terms of the flux of the electric field through the surface itself. Flux is the integral of a differential form and, consequently, a

de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...

class. It is not by chance that formulas of this kind, such as the well known Stokes formula, though being a natural part of classical differential calculus, have entered in modern mathematics from physics.

Classical analogues

All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differentiable manifolds. The Euler operator, which associates to each

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 ...

the corresponding

Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order. More generally, the analog of

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...

s in secondary calculus are the elements of the first term of the so-called C-spectral sequence, and so on. The simplest diffieties are infinite prolongations of partial differential equations, which are subvarieties of infinite jet spaces. The latter are infinite dimensional varieties that can not be studied by means of standard

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

. On the contrary, the most natural language in which to study these objects is

differential calculus over commutative algebras In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of thi ...

. Therefore, the latter must be regarded as a fundamental tool of secondary calculus. On the other hand, differential calculus over commutative algebras gives the possibility to develop algebraic geometry as if it were differential geometry.

Theoretical physics

Recent developments of

particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and ...

, based on quantum field theories and its generalizations, have led to understand the deep cohomological nature of the quantities describing both classical and quantum fields. The turning point was the discovery of the famous BRST transformation. For instance, it was understood that observables in field theory are classes in horizontal de Rham cohomology which are invariant under the corresponding gauge group and so on. This current in modern theoretical physics is actually growing and it is called Cohomological Physics. It is relevant that secondary calculus and cohomological physics, which developed for twenty years independently from each other, arrived at the same results. Their confluence took place at the international conference ''Secondary Calculus and Cohomological Physics'' (Moscow, August 24–30, 1997).

Prospects

A large number of modern mathematical theories harmoniously converges in the framework of secondary calculus, for instance:

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

and algebraic geometry,

homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology ...

and

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

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 addit ...

and

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 iden ...

theory, differential geometry, etc.

References

* I. S. Krasil'shchik, ''Calculus over Commutative Algebras: a concise user's guide'', Acta Appl. Math. 49 (1997) 235—248
DIPS-01/98
* I. S. Krasil'shchik, A. M. Verbovetsky, ''Homological Methods in Equations of Mathematical Physics'', Open Ed. and Sciences, Opava (Czech Rep.), 1998

* I. S. Krasil'shchik, A. M. Vinogradov (eds.), ''Symmetries and conservation laws for differential equations of mathematical physics'', Translations of Math. Monographs 182, Amer. Math. Soc., 1999. * J. Nestruev, ''Smooth Manifolds and Observables'', Graduate Texts in Mathematics 220, Springer, 2002, . * A. M. Vinogradov, ''The C-spectral sequence, Lagrangian formalism, and conservation laws I. The linear theory'', J. Math. Anal. Appl. 100 (1984) 1—40
Diffiety Inst. Library
* A. M. Vinogradov, ''The C-spectral sequence, Lagrangian formalism, and conservation laws II. The nonlinear theory'', J. Math. Anal. Appl. 100 (1984) 41—129
Diffiety Inst. Library
* A. M. Vinogradov, ''From symmetries of partial differential equations towards secondary (`quantized') calculus'', J. Geom. Phys. 14 (1994) 146—194
Diffiety Inst. Library
* A. M. Vinogradov, ''Introduction to Secondary Calculus'', Proc. Conf. Secondary Calculus and Cohomology Physics (M. Henneaux, I. S. Krasil'shchik, and A. M. Vinogradov, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, Rhode Island, 1998

* A. M. Vinogradov, ''Cohomological Analysis of Partial Differential Equations and Secondary Calculus'', Translations of Math. Monographs 204, Amer. Math. Soc., 2001.

External links

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