In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a line field on a
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 N ...
is a formation of a
line being tangent to a manifold at each point, i.e. a
section
Section, Sectioning, or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of the
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
over the manifold. Line fields are of particular interest in the study of
complex dynamical systems, where it is conventional to modify the definition slightly.
Definitions
In general, let ''M'' be a manifold. A line field on ''M'' is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
''μ'' that assigns to each point ''p'' of ''M'' a line ''μ''(''p'') through the origin in the tangent space T
''p''(''M''). Equivalently, one may say that ''μ''(''p'') is an element of the
projective tangent space PT
''p''(''M''), or that ''μ'' is a section of the projective
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
PT(''M'').
In the study of complex dynamical systems, the manifold ''M'' is taken to be a
Hersee surface. A line field on a subset ''A'' of ''M'' (where ''A'' is required to have positive two-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
) is a line field on ''A'' in the general sense above that is defined
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
in ''A'' and is also a
measurable function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
.
References
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