Editing Mathematical analysis (section)

== Main branches ==

=== Calculus ===
{{Main|Calculus}}

=== Real analysis ===
{{Main|Real analysis}}
Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the [[real number]]s and real-valued functions of a real variable.<ref>{{cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Principles of Mathematical Analysis |url=https://archive.org/details/principlesofmath00rudi |url-access=registration |series=Walter Rudin Student Series in Advanced Mathematics |date=1976 |edition=3rd |publisher=McGraw–Hill |isbn=978-0070542358}}</ref><ref>{{cite book |last=Abbott |first=Stephen |title=Understanding Analysis |series=Undergraduate Texts in Mathematics |isbn=978-0387950600 |date=2001 |location=New York |publisher=Springer-Verlag}}</ref> In particular, it deals with the analytic properties of real [[function (mathematics)|functions]] and [[sequence]]s, including [[Limit of a sequence|convergence]] and [[limit of a function|limits]] of [[sequence]]s of real numbers, the [[calculus]] of the real numbers, and [[continuous function|continuity]], [[smooth function|smoothness]] and related properties of real-valued functions.

=== Complex analysis ===
{{Main|Complex analysis}}

Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates [[Function (mathematics)|functions]] of [[complex numbers]].<ref name="Ahlfors_1979"/> It is useful in many branches of mathematics, including [[algebraic geometry]], [[number theory]], [[applied mathematics]]; as well as in [[physics]], including [[hydrodynamics]], [[thermodynamics]], [[mechanical engineering]], [[electrical engineering]], and particularly, [[quantum field theory]].

Complex analysis is particularly concerned with the [[analytic function]]s of complex variables (or, more generally, [[meromorphic function]]s). Because the separate [[real number|real]] and [[imaginary number|imaginary]] parts of any analytic function must satisfy [[Laplace's equation]], complex analysis is widely applicable to two-dimensional problems in [[physics]].

=== Functional analysis ===
{{Main|Functional analysis}}
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of [[vector space]]s endowed with some kind of limit-related structure (e.g. [[Inner product space#Definition|inner product]], [[Norm (mathematics)#Definition|norm]], [[Topological space#Definitions|topology]], etc.) and the [[linear transformation|linear operators]] acting upon these spaces and respecting these structures in a suitable sense.<ref name="Rudin_1991"/><ref name="Conway_1994"/> The historical roots of functional analysis lie in the study of [[function space|spaces of functions]] and the formulation of properties of transformations of functions such as the [[Fourier transform]] as transformations defining [[continuous function|continuous]], [[unitary operator|unitary]] etc. operators between function spaces.  This point of view turned out to be particularly useful for the study of [[differential equations|differential]] and [[integral equations]].

=== Harmonic analysis ===
{{Main|Harmonic analysis}}
Harmonic analysis is a branch of mathematical analysis concerned with the representation of [[function (mathematics)|function]]s and [[signal]]s as the superposition of basic [[wave]]s. This includes the study of the notions of [[Fourier series]] and [[Fourier transform]]s ([[Fourier analysis]]), and of their generalizations. Harmonic analysis has applications in areas as diverse as [[music theory]], [[number theory]], [[representation theory]], [[signal processing]], [[quantum mechanics]], [[tidal analysis]], and [[neuroscience]].

=== Differential equations ===
{{Main|Differential equation}}
A differential equation is a [[mathematics|mathematical]] [[equation]] for an unknown [[function (mathematics)|function]] of one or several [[Variable (mathematics)|variables]] that relates the values of the function itself and its [[derivative]]s of various [[Derivative#Higher derivatives|orders]].<ref>{{cite book|first = Edward L.|last =  Ince|title =Ordinary Differential Equations|publisher =  Dover Publications|date = 1956|isbn=978-0486603490|url = https://books.google.com/books?id=mbyqAAAAQBAJ}}</ref><ref>[[Witold Hurewicz]], ''Lectures on Ordinary Differential Equations'', Dover Publications, {{isbn|0486495108}}</ref><ref name="Evans_1998"/> Differential equations play a prominent role in [[engineering]], [[physics]], [[economics]], [[biology]], and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a [[Deterministic system (mathematics)|deterministic]] relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in [[classical mechanics]], where the motion of a body is described by its position and velocity as the time value varies. [[Newton's laws of motion|Newton's laws]] allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an [[equations of motion|equation of motion]]) may be solved explicitly.

=== Measure theory ===
{{Main|Measure (mathematics)}}

A measure on a [[set (mathematics)|set]] is a systematic way to assign a number to each suitable [[subset]] of that set, intuitively interpreted as its size.<ref>{{cite book|author-link = Terence Tao|first = Terence|last = Tao|date = 2011|title = An Introduction to Measure Theory| series=Graduate Studies in Mathematics | volume=126 |publisher = American Mathematical Society|isbn = 978-0821869192|url = https://books.google.com/books?id=HoGDAwAAQBAJ|access-date = 2018-10-26|archive-date = 2019-12-27|archive-url = https://web.archive.org/web/20191227145317/https://books.google.com/books?id=HoGDAwAAQBAJ|url-status = live|doi=10.1090/gsm/126}}</ref> In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the [[Lebesgue measure]] on a [[Euclidean space]], which assigns the conventional [[length]], [[area]], and [[volume]] of [[Euclidean geometry]] to suitable subsets of the <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math>. For instance, the Lebesgue measure of the [[Interval (mathematics)|interval]] <math>\left[0, 1\right]</math> in the [[real line|real numbers]] is its length in the everyday sense of the word – specifically, 1.

Technically, a measure is a function that assigns a non-negative real number or [[Extended real number line|+∞]] to (certain) subsets of a set <math>X</math>. It must assign 0 to the [[empty set]] and be ([[countably]]) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a ''consistent'' size to ''each'' subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the [[counting measure]]. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called ''measurable'' subsets, which are required to form a [[Sigma-algebra|<math>\sigma</math>-algebra]]. This means that the empty set, countable [[union (set theory)|unions]], countable [[intersection (set theory)|intersections]] and [[complement (set theory)|complements]] of measurable subsets are measurable. [[Non-measurable set]]s in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the [[axiom of choice]].

=== Numerical analysis ===
{{Main|Numerical analysis}}
Numerical analysis is the study of [[algorithm]]s that use numerical [[approximation]] (as opposed to general [[symbolic computation|symbolic manipulations]]) for the problems of mathematical analysis (as distinguished from [[discrete mathematics]]).<ref>{{cite book |last=Hildebrand |first=Francis B. | author-link=Francis B. Hildebrand | title=Introduction to Numerical Analysis | edition=2nd |date=1974 |publisher=McGraw-Hill |isbn= 978-0070287617}}</ref>

Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st&nbsp;century, the life sciences and even the arts have adopted elements of scientific computations. [[Ordinary differential equation]]s appear in [[celestial mechanics]] (planets, stars and galaxies); [[numerical linear algebra]] is important for data analysis; [[stochastic differential equation]]s and [[Markov chain]]s are essential in simulating living cells for medicine and biology.

=== Vector analysis ===
{{Main|Vector calculus}}
{{See also|A History of Vector Analysis|Vector Analysis}}
''Vector analysis'', also called ''vector calculus'', is a branch of mathematical analysis dealing with [[vector-valued function]]s.<ref>{{cite book |last1=Borisenko |first1=A. I. |last2=Tarapov |first2=I. E. |title=Vector and Tensor Analysis with Applications (Dover Books on Mathematics) |date=1979 |publisher=Dover Books on Mathematics}}</ref>

=== Scalar analysis ===
{{Main|Scalar (mathematics)}}
Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.

=== Tensor analysis ===
{{Main|Tensor field}}