In mathematics, topological degree theory is a generalization of the
winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...
of a curve in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. It can be used to estimate the number of solutions of an equation, and is closely connected to
fixed-point theory Fixed point may refer to:
* Fixed point (mathematics), a value that does not change under a given transformation
* Fixed-point arithmetic, a manner of doing arithmetic on computers
* Fixed point, a benchmark (surveying) used by geodesists
* Fixed p ...
. When one solution of an equation is easily found, degree theory can often be used to prove existence of a second, nontrivial, solution. There are different types of degree for different types of maps: e.g. for maps between
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
there is the
Brouwer degree
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree ...
in R
''n'', the
Leray-Schauder degree for
compact mappings in
normed spaces
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
, the
coincidence degree
A coincidence is a remarkable concurrence of events or circumstances that have no apparent causal connection with one another. The perception of remarkable coincidences may lead to supernatural, occult, or paranormal claims, or it may lead to ...
and various other types. There is also a degree for
continuous maps between manifolds.
Topological degree theory has applications in
complementarity problem
A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements (constraints) which include: that the inner pro ...
s,
differential equations
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, an ...
,
differential inclusion
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form
:\frac(t)\in F(t,x(t)),
where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point ...
s and
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s.
Further reading
Topological fixed point theory of multivalued mappings Lech Górniewicz, Springer, 1999,
Topological degree theory and applications Donal O'Regan, Yeol Je Cho, Yu Qing Chen, CRC Press, 2006,
Mapping Degree Theory Enrique Outerelo, Jesus M. Ruiz, AMS Bookstore, 2009,
{{topology-stub
Topology
Algebraic topology
Differential topology