Editing Mathematical analysis

{{short description|Branch of mathematics}}
{{use dmy dates|date=May 2021|cs1-dates=y}}
{{Math topics TOC}}

[[File:Attracteur étrange de Lorenz.png|thumb|upright=1.2|A [[strange attractor]] arising from a [[differential equation]]. Differential equations are an important area of mathematical analysis with many applications in science and engineering.]]
'''Analysis''' is the branch of [[mathematics]] dealing with [[continuous function]]s, [[limit (mathematics)|limit]]s, and related theories, such as [[Derivative|differentiation]], [[Integral|integration]], [[measure (mathematics)|measure]], [[infinite sequence]]s, [[series (mathematics)|series]], and [[analytic function]]s.<ref>[[Edwin Hewitt]] and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965</ref><ref name="Stillwell_Analysis"/>

These theories are usually studied in the context of [[Real number|real]] and [[Complex number|complex]] numbers and [[Function (mathematics)|functions]]. Analysis evolved from [[calculus]], which involves the elementary concepts and techniques of analysis.
Analysis may be distinguished from [[geometry]]; however, it can be applied to any [[Space (mathematics)|space]] of [[mathematical object]]s that has a definition of nearness (a [[topological space]]) or specific distances between objects  (a [[metric space]]).

== History ==
[[Image:Archimedes pi.svg|thumb|right|300px|[[Archimedes]] used the [[method of exhaustion]] to compute the [[area]] inside a circle by finding the area of [[regular polygon]]s with more and more sides. This was an early but informal example of a [[limit (mathematics)|limit]], one of the most basic concepts in mathematical analysis.]]

===Ancient===
Mathematical analysis formally developed in the 17th century during the [[Scientific Revolution]],<ref name=analysis>{{cite book|last=Jahnke|first=Hans Niels|title=A History of Analysis|series=History of Mathematics |url=https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7|date=2003|volume=24 |publisher=[[American Mathematical Society]]|isbn=978-0821826232|page=7|access-date=2015-11-15|archive-date=2016-05-17|archive-url=https://web.archive.org/web/20160517180439/https://books.google.com/books?id=CVRZEXFVsZkC&pg=PR7|url-status=live|doi=10.1090/hmath/024}}</ref> but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of [[Greek mathematics|ancient Greek mathematics]]. For instance, an [[geometric series|infinite geometric sum]] is implicit in [[Zeno of Elea|Zeno's]] [[Zeno's paradoxes#Dichotomy paradox|paradox of the dichotomy]].<ref name="Stillwell_2004"/> (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, [[Greek mathematics|Greek mathematicians]] such as [[Eudoxus of Cnidus|Eudoxus]] and [[Archimedes]] made more explicit, but informal, use of the concepts of limits and convergence when they used the [[method of exhaustion]] to compute the area and volume of regions and solids.<ref name="Smith_1958"/> The explicit use of [[infinitesimals]] appears in Archimedes' ''[[The Method of Mechanical Theorems]]'', a work rediscovered in the 20th century.<ref>{{cite book|last=Pinto|first=J. Sousa|title=Infinitesimal Methods of Mathematical Analysis|url=https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7|date=2004|publisher=Horwood Publishing|isbn=978-1898563990|page=8|access-date=2015-11-15|archive-date=2016-06-11|archive-url=https://web.archive.org/web/20160611045431/https://books.google.com/books?id=bLbfhYrhyJUC&pg=PA7|url-status=live}}</ref> In Asia, the [[Chinese mathematics|Chinese mathematician]] [[Liu Hui]] used the method of exhaustion in the 3rd century CE to find the area of a circle.<ref>{{cite book|series=Chinese studies in the history and philosophy of science and technology|volume=130|title=A comparison of Archimedes' and Liu Hui's studies of circles|first1=Liu|last1=Dun|first2=Dainian|last2=Fan|first3=Robert Sonné|last3=Cohen|publisher=Springer|date=1966|isbn=978-0-7923-3463-7|page=279|url=https://books.google.com/books?id=jaQH6_8Ju-MC|access-date=2015-11-15|archive-date=2016-06-17|archive-url=https://web.archive.org/web/20160617055211/https://books.google.com/books?id=jaQH6_8Ju-MC|url-status=live}}, [https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 Chapter, p. 279] {{Webarchive|url=https://web.archive.org/web/20160526221958/https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 |date=2016-05-26 }}</ref> From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the [[arithmetic series|arithmetic]] and [[geometric series|geometric]] series as early as the 4th century BCE.<ref>{{cite journal | title = On the Use of Series in Hindu Mathematics | author = Singh, A. N. | journal  =    Osiris | volume = 1  |date = 1936  | pages = 606–628 | doi = 10.1086/368443 | jstor = 301627 | s2cid = 144760421 | url = https://www.jstor.org/stable/301627}}</ref>
[[Bhadrabahu|Ācārya Bhadrabāhu]] uses the sum of a geometric series in his Kalpasūtra in {{BCE|433}}.<ref>{{cite journal | title = Summation of Convergent Geometric Series and the concept of approachable Sunya | author = K. B. Basant, Satyananda Panda | journal = Indian Journal of History of Science | volume = 48  |date = 2013  | pages = 291–313 | url = https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol48_2_7_KBBasant.pdf}}</ref>

===Medieval===
[[Zu Chongzhi]] established a method that would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]] in the 5th century.<ref>{{cite book|title=Calculus: Early Transcendentals|edition=3|first1=Dennis G.|last1=Zill|first2=Scott|last2=Wright|first3=Warren S.|last3=Wright|publisher=Jones & Bartlett Learning|date=2009|isbn=978-0763759957|page=xxvii|url=https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27|access-date=2015-11-15|archive-date=2019-04-21|archive-url=https://web.archive.org/web/20190421114230/https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27|url-status=live}}</ref> In the 12th century, the [[Indian mathematics|Indian mathematician]] [[Bhāskara II]] used infinitesimal and used what is now known as [[Rolle's theorem]].<ref>{{citation|title=The positive sciences of the ancient Hindus|journal=Nature|volume=97|issue=2426|page=177|first=Sir Brajendranath|last=Seal|date=1915|bibcode=1916Natur..97..177.|doi=10.1038/097177a0|hdl=2027/mdp.39015004845684|s2cid=3958488|hdl-access=free}}</ref>

In the 14th century, [[Madhava of Sangamagrama]] developed [[series (mathematics)|infinite series]] expansions, now called [[Taylor series]], of functions such as [[Trigonometric functions|sine]], [[Trigonometric functions|cosine]], [[trigonometric functions|tangent]] and [[Inverse trigonometric functions|arctangent]].<ref name=rajag78>
{{cite journal
 | title =       On an untapped source of medieval Keralese Mathematics
 | first1=      C. T.
 | last1=      Rajagopal
 | first2 =      M. S.
 | last2=      Rangachari
 | journal  =    Archive for History of Exact Sciences
 | volume =      18 | number=2
 |date=June 1978
 | pages =       89–102 | doi = 10.1007/BF00348142
| s2cid= 51861422
 }}</ref> Alongside his development of Taylor series of [[trigonometric functions]], he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series.  His followers at the [[Kerala School of Astronomy and Mathematics]] further expanded his works, up to the 16th century.

===Modern===
====Foundations====
The modern foundations of mathematical analysis were established in 17th century Europe.<ref name=analysis/> This began when [[Fermat]] and [[Descartes]] developed [[analytic geometry]], which is the precursor to modern calculus. Fermat's method of [[adequality]] allowed him to determine the maxima and minima of functions and the tangents of curves.<ref name=Pellegrino>{{cite web | last = Pellegrino | first = Dana | title = Pierre de Fermat | url = http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | access-date = 2008-02-24 | archive-date = 2008-10-12 | archive-url = https://web.archive.org/web/20081012024028/http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | url-status = live }}</ref> Descartes's publication of ''[[La Géométrie]]'' in 1637, which introduced the [[Cartesian coordinate system]], is considered to be the establishment of mathematical analysis. It would be a few decades later that [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] independently developed [[infinitesimal calculus]], which grew, with the stimulus of applied work that continued through the 18th&nbsp;century, into analysis topics such as the [[calculus of variations]], [[Ordinary differential equation|ordinary]] and [[partial differential equation]]s, [[Fourier analysis]], and [[generating function]]s. During this period, calculus techniques were applied to approximate [[discrete mathematics|discrete problems]] by continuous ones.

====Modernization====
In the 18th century, [[Leonhard Euler|Euler]] introduced the notion of a [[function (mathematics)|mathematical function]].<ref name="function">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All| url = https://archive.org/details/eulermasterofusa0000dunh| url-access = registration| date = 1999| publisher =The Mathematical Association of America | page= [https://archive.org/details/eulermasterofusa0000dunh/page/17 17]}}</ref> Real analysis began to emerge as an independent subject when [[Bernard Bolzano]] introduced the modern definition of continuity in 1816,<ref>*{{cite book |first=Roger |last=Cooke |author-link=Roger Cooke (mathematician) |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |date=1997 |isbn=978-0471180821 |page=[https://archive.org/details/historyofmathema0000cook/page/379 379] |chapter=Beyond the Calculus |quote=Real analysis began its growth as an independent subject with the introduction of the modern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781–1848) |chapter-url=https://archive.org/details/historyofmathema0000cook/page/379 }}</ref> but Bolzano's work did not become widely known until the 1870s. In 1821, [[Augustin Louis Cauchy|Cauchy]] began to put calculus on a firm logical foundation by rejecting the principle of the [[generality of algebra]] widely used in earlier work, particularly by Euler.  Instead, Cauchy formulated calculus in terms of geometric ideas and [[infinitesimal]]s.  Thus, his definition of continuity required an infinitesimal change in ''x'' to correspond to an infinitesimal change in ''y''.  He also introduced the concept of the [[Cauchy sequence]], and started the formal theory of [[complex analysis]]. [[Siméon Denis Poisson|Poisson]], [[Joseph Liouville|Liouville]], [[Joseph Fourier|Fourier]] and others studied partial differential equations and [[harmonic analysis]].  The contributions of these mathematicians and others, such as [[Karl Weierstrass|Weierstrass]], developed the [[(ε, δ)-definition of limit]] approach, thus founding the modern field of mathematical analysis. Around the same time, [[Bernhard Riemann|Riemann]] introduced his theory of [[integral|integration]], and made significant advances in complex analysis.

Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a [[Continuum (set theory)|continuum]] of [[real number]]s without proof. [[Richard Dedekind|Dedekind]] then constructed the real numbers by [[Dedekind cut]]s, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a [[complete metric space|complete]] set: the continuum of real numbers, which had already been developed by [[Simon Stevin]] in terms of [[decimal expansion]]s. Around that time, the attempts to refine the [[theorem]]s of [[Riemann integral|Riemann integration]] led to the study of the "size" of the set of [[Classification of discontinuities|discontinuities]] of real functions.

Also, various [[pathological (mathematics)|pathological objects]], (such as [[nowhere continuous function]]s, continuous but [[Weierstrass function|nowhere differentiable functions]], and [[space-filling curve]]s), commonly known as "monsters", began to be investigated. In this context, [[Camille Jordan|Jordan]] developed his theory of [[Jordan measure|measure]], [[Georg Cantor|Cantor]] developed what is now called [[naive set theory]], and [[René-Louis Baire|Baire]] proved the [[Baire category theorem]]. In the early 20th century, calculus was formalized using an axiomatic [[set theory]]. [[Henri Lebesgue|Lebesgue]] greatly improved measure theory, and introduced his own theory of integration, now known as [[Lebesgue integration]], which proved to be a big improvement over Riemann's. [[David Hilbert|Hilbert]] introduced [[Hilbert space]]s to solve [[integral equation]]s. The idea of [[normed vector space]] was in the air, and in the 1920s [[Stefan Banach|Banach]] created [[functional analysis]].

== Important concepts ==
=== Metric spaces ===
{{Main|Metric space}}
In [[mathematics]], a metric space is a [[Set (mathematics)|set]] where a notion of [[distance]] (called a [[metric (mathematics)|metric]]) between elements of the set is defined.

Much of analysis happens in some metric space; the most commonly used are the [[real line]], the [[complex plane]], [[Euclidean space]], other [[vector space]]s, and the [[integer]]s. Examples of analysis without a metric include [[measure theory]] (which describes size rather than distance) and [[functional analysis]] (which studies [[topological vector space]]s that need not have any sense of distance).

Formally, a metric space is an [[ordered pair]] <math>(M,d)</math> where <math>M</math> is a set and <math>d</math> is a [[metric (mathematics)|metric]] on <math>M</math>, i.e., a [[Function (mathematics)|function]]

:<math>d \colon M \times M \rightarrow \mathbb{R}</math>

such that for any <math>x, y, z \in M</math>, the following holds:

# <math>d(x,y) \geq 0</math>, with equality [[if and only if]] <math>x = y</math> &nbsp;&nbsp; (''[[identity of indiscernibles]]''),
# <math>d(x,y) = d(y,x)</math> &nbsp;&nbsp; (''symmetry''), and
# <math>d(x,z) \le d(x,y) + d(y,z)</math> &nbsp;&nbsp; (''[[triangle inequality]]'').

By taking the third property and letting <math>z=x</math>, it can be shown that <math>d(x,y) \ge 0</math> &nbsp;&nbsp;&nbsp; (''non-negative'').

=== Sequences and limits ===
{{Main|Sequence}}
{{See also|Limit of a sequence}}
A sequence is an ordered list. Like a [[Set (mathematics)|set]], it contains [[Element (mathematics)|members]] (also called ''elements'', or ''terms''). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a [[function (mathematics)|function]] whose domain is a [[countable]] [[totally ordered]] set, such as the [[natural numbers]].

One of the most important properties of a sequence is ''convergence''. Informally, a sequence converges if it has a ''limit''. Continuing informally, a ([[#Finite and infinite|singly-infinite]]) sequence has a limit if it approaches some point ''x'', called the limit, as ''n'' becomes very large. That is, for an abstract sequence (''a''<sub>''n''</sub>) (with ''n'' running from 1 to infinity understood) the distance between ''a''<sub>''n''</sub> and ''x'' approaches 0 as ''n'' → ∞, denoted
:<math>\lim_{n\to\infty} a_n = x.</math>

== 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}}

== Other topics ==
* [[Calculus of variations]] deals with extremizing [[functional (mathematics)|functionals]], as opposed to ordinary [[calculus]] which deals with [[function (mathematics)|functions]].
* [[Harmonic analysis]] deals with the representation of [[function (mathematics)|functions]] or signals as the [[superposition principle|superposition]] of basic [[wave]]s.
* [[Geometric analysis]] involves the use of geometrical methods in the study of [[partial differential equation]]s and the application of the theory of partial differential equations to geometry.
* [[Clifford analysis]], the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators,  termed in general as monogenic or Clifford analytic functions.
* [[p-adic analysis|''p''-adic analysis]], the study of analysis within the context of [[p-adic number|''p''-adic numbers]], which differs in some interesting and surprising ways from its real and complex counterparts.
* [[Non-standard analysis]], which investigates the [[hyperreal number]]s and their functions and gives a [[rigour#Mathematical rigour|rigorous]] treatment of [[infinitesimal]]s and infinitely large numbers.
* [[Computable analysis]], the study of which parts of analysis can be carried out in a [[computability theory|computable]] manner.
* [[Stochastic calculus]] – analytical notions developed for [[stochastic processes]].
* [[Set-valued analysis]] – applies ideas from analysis and topology to set-valued functions.
* [[Convex analysis]], the study of convex sets and functions.
* [[Idempotent analysis]] – analysis in the context of an [[idempotent semiring]], where the lack of an additive inverse is compensated somewhat by the idempotent rule A&nbsp;+&nbsp;A&nbsp;=&nbsp;A.
** [[Tropical analysis]]  – analysis of the idempotent semiring called the [[tropical semiring]] (or [[max-plus algebra]]/[[min-plus algebra]]).
* [[Constructive analysis]], which is built upon a foundation of [[constructive logic|constructive]], rather than classical, logic and set theory.
* [[Intuitionistic analysis]], which is developed from constructive logic like constructive analysis but also incorporates [[choice sequence]]s.
* [[Paraconsistent analysis]], which is built upon a foundation of [[paraconsistent logic|paraconsistent]], rather than classical, logic and set theory.
* [[Smooth infinitesimal analysis]], which is developed in a smooth topos.

== Applications ==
Techniques from analysis are also found in other areas such as:

=== Physical sciences ===
The vast majority of [[classical mechanics]], [[Theory of relativity|relativity]], and [[quantum mechanics]] is based on applied analysis, and [[differential equation]]s in particular. Examples of important differential equations include [[Newton's second law]], the [[Schrödinger equation]], and the [[Einstein field equations]].

[[Functional analysis]] is also a major factor in [[quantum mechanics]].

=== Signal processing ===
When processing signals, such as [[Sound|audio]], [[radio wave]]s, light waves, [[seismic waves]], and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal.  A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.<ref>{{cite book |title=Theory and Application of Digital Signal Processing |last1=Rabiner |first1=L. R. |last2=Gold |first2=B. |location=Englewood Cliffs, New Jersey |publisher=[[Prentice-Hall]] |date=1975 |isbn=978-0139141010 |url=https://archive.org/details/theoryapplicatio00rabi |url-access=registration}}</ref>

=== Other areas of mathematics ===
Techniques from analysis are used in many areas of mathematics, including:
* [[Analytic number theory]]
* [[Analytic combinatorics]]
* [[Continuous probability]]
* [[Differential entropy]] in information theory
* [[Differential game]]s
* [[Differential geometry]], the application of calculus to specific mathematical spaces known as [[manifold]]s that possess a complicated internal structure but behave in a simple manner locally.
* [[Differentiable manifolds]]
* [[Differential topology]]
* [[Partial differential equations]]

== Famous Textbooks ==
* Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau 
* Introductory Real Analysis, by [[Andrey Kolmogorov]], [[Sergei Fomin]]<ref>{{cite web | url=https://archive.org/details/kolmogorov-fomin-introductory-real-analysis | title=Introductory Real Analysis | year=1970 }}</ref>
* Differential and Integral Calculus (3 volumes), by [[Grigorii Fichtenholz]]<ref>{{cite web | url=https://archive.org/details/B-001-014-344/mode/2up | title=Курс дифференциального и интегрального исчисления. Том I | year=1969 }}</ref><ref>{{cite web | url=https://archive.org/details/B-001-014-359 | title=Основы математического анализа. Том II | year=1960 }}</ref><ref>{{cite web | url=https://archive.org/details/B-001-014-346 | title=Курс дифференциального и интегрального исчисления. Том III | year=1960 }}</ref>
* The Fundamentals of Mathematical Analysis (2 volumes), by [[Grigorii Fichtenholz]]<ref>{{cite book |title=The Fundamentals of Mathematical Analysis: International Series in Pure and Applied Mathematics, Volume 1 | id={{ASIN|0080134734|country=ca}} }}</ref><ref>{{cite book |title=The Fundamentals of Mathematical Analysis: International Series of Monographs in Pure and Applied Mathematics, Vol. 73-II | id={{ASIN|1483213153|country=ca}} }}</ref>
* A Course Of Mathematical Analysis (2 volumes), by [[Sergey Nikolsky]]<ref>{{cite web | url=https://archive.org/details/nikolsky-a-course-of-mathematical-analysis-vol-1-mir/page/1/mode/2up | title=A Course of Mathematical Analysis Vol 1 | year=1977 }}</ref><ref>{{cite web | url=https://archive.org/details/nikolsky-a-course-of-mathematical-analysis-vol-2-mir | title=A Course of Mathematical Analysis Vol 2 | year=1987 }}</ref>
* Mathematical Analysis (2 volumes), by [[Vladimir A. Zorich|Vladimir Zorich]]<ref>{{cite book |title=Mathematical Analysis I | id={{ASIN|3662569558|country=ca}} }}</ref><ref>{{cite book |title=Mathematical Analysis II | id={{ASIN|3662569663|country=ca}} }}</ref>
* A Course of Higher Mathematics (5 volumes, 6 parts), by [[Vladimir Smirnov (mathematician)|Vladimir Smirnov]]<ref name="archive.org">{{cite web | url=https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-3-1-linear-algebra | title=A Course of Higher Mathematics Vol 3 1 Linear Algebra | year=1964 }}</ref><ref>{{cite web | url=https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-2-advanced-calculus | title=A Course of Higher Mathematics Vol 2 Advanced Calculus | year=1964 }}</ref><ref>{{cite web | url=https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-3-2-complex-variables-special-functions | title=A Course of Higher Mathematics Vol 3-2 Complex Variables Special Functions | year=1964 }}</ref><ref>{{cite web | url=https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-4-integral-and-partial-differential-equations | title=A Course of Higher Mathematics Vol 4 Integral and Partial Differential Equations | year=1964 }}</ref><ref>{{cite web | url=https://archive.org/details/smirnov-a-course-of-higher-mathematics-vol-5-integration-and-functional-analysis | title=A Course of Higher Mathematics Vol 5 Integration and Functional Analysis | year=1964 }}</ref>
* Differential And Integral Calculus, by [[Nikolai Piskunov]]<ref>{{cite web | url=https://archive.org/details/n.-piskunov-differential-and-integral-calculus-mir-1969/page/1/mode/2up | title=Differential and Integral Calculus | year=1969 }}</ref>
* A Course of Mathematical Analysis, by [[Aleksandr Khinchin]]<ref>{{cite web | url=https://archive.org/details/khinchin-a-course-of-mathematical-analysis | title=A Course of Mathematical Analysis | year=1960 }}</ref>
* Mathematical Analysis: A Special Course, by [[Georgiy Shilov]]<ref>{{cite book |title=Mathematical Analysis: A Special Course | id={{ASIN|1483169561|country=ca}} }}</ref>
* Theory of Functions of a Real Variable (2 volumes), by [[Isidor Natanson]]<ref>{{cite web | url=https://archive.org/details/theoryoffunction00nata | title=Theory of functions of a real variable (Teoria functsiy veshchestvennoy peremennoy, chapters I to IX) | year=1955 }}</ref><ref>{{cite web | url=https://archive.org/details/theoryoffunction0002nata | title=Theory of functions of a real variable =Teoria functsiy veshchestvennoy peremennoy | year=1955 }}</ref>
* Problems in Mathematical Analysis, by [[Boris Demidovich]]<ref>{{cite web | url=https://archive.org/details/DemidovichEtAlProblemsInMathematicalAnalysisMir1970 | title=Problems in Mathematical Analysis | year=1970 }}</ref>
* [[Problems and Theorems in Analysis]] (2 volumes), by [[George Pólya]], [[Gábor Szegő]]<ref>{{cite book |title=Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions | id={{ASIN|3540636404|country=ca}} }}</ref><ref>{{cite book |title=Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry | id={{ASIN|3540636862|country=ca}} }}</ref>
* Mathematical Analysis: A Modern Approach to Advanced Calculus, by [[Tom M. Apostol|Tom Apostol]]<ref>{{cite book |title=Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd Edition | id={{ASIN|0201002884|country=ca}} }}</ref>
* Principles of Mathematical Analysis, by [[Walter Rudin]]<ref>{{cite book |title=Principles of Mathematical Analysis | id={{ASIN|0070856133|country=ca}} }}</ref>
* Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by [[Elias M. Stein|Elias Stein]]<ref>{{cite book |title=Real Analysis: Measure Theory, Integration, and Hilbert Spaces | id={{ASIN|0691113866|country=ca}} }}</ref>
* Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by [[Lars Ahlfors]]<ref>{{cite book|title=Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable|isbn=978-0070006577|date=January 1, 1979 |last1=Ahlfors |first1=Lars |publisher=McGraw-Hill Education }}</ref>
* Complex Analysis, by [[Elias M. Stein|Elias Stein]]<ref>{{cite book |title=Complex Analysis | id={{ASIN|0691113858|country=ca}} }}</ref>
* Functional Analysis: Introduction to Further Topics in Analysis, by [[Elias M. Stein|Elias Stein]]<ref>{{cite book |title=Functional Analysis: Introduction to Further Topics in Analysis | id={{ASIN|0691113874|country=ca}} }}</ref>
* Analysis (2 volumes), by [[Terence Tao]]<ref>{{cite book |title=Analysis I: Third Edition | id={{ASIN|9380250649|country=ca}} }}</ref><ref>{{cite book |title=Analysis II: Third Edition | id={{ASIN|9380250657|country=ca}} }}</ref>
* Analysis (3 volumes), by Herbert Amann, Joachim Escher<ref>{{cite book |isbn=978-3764371531|title=Analysis I |last1=Amann |first1=Herbert |last2=Escher |first2=Joachim |date= 2004 |publisher=Birkhäuser }}</ref><ref>{{cite book |isbn=978-3764374723|title=Analysis II |last1=Amann |first1=Herbert |last2=Escher |first2=Joachim |date=16 May 2008 |publisher=Birkhäuser Basel }}</ref><ref>{{cite book |isbn=978-3764374792|title=Analysis III |last1=Amann |first1=Herbert |last2=Escher |first2=Joachim |date= 2009 |publisher=Springer }}</ref>
* Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov<ref>{{cite book |isbn=978-3030382216|title=Real and Functional Analysis |last1=Bogachev |first1=Vladimir I. |last2=Smolyanov |first2=Oleg G. |date= 2021 |publisher=Springer }}</ref>
* Real and Functional Analysis, by [[Serge Lang]]<ref>{{cite book |isbn=978-1461269380|title=Real and Functional Analysis |last1=Lang |first1=Serge |date= 2012 |publisher=Springer }}</ref>

== See also ==
{{Portal|Mathematics}}
* [[Constructive analysis]]
* [[History of calculus]]
* [[Hypercomplex analysis]]
* [[Multiple rule-based problems]]
* [[Multivariable calculus]]
* [[Paraconsistent logic]]
* [[Smooth infinitesimal analysis]]
* [[Timeline of calculus and mathematical analysis]]

== References ==
{{reflist|refs=
<ref name="Stillwell_Analysis">{{cite encyclopedia |title=analysis {{!}} mathematics |url=https://www.britannica.com/topic/analysis-mathematics |access-date=2015-07-31 |encyclopedia=Encyclopædia Britannica |author-first=John Colin |author-last=Stillwell |author-link=John Colin Stillwell |date= |archive-date=2015-07-26 |archive-url=https://web.archive.org/web/20150726223522/https://www.britannica.com/topic/analysis-mathematics |url-status=live}}</ref>
<ref name="Stillwell_2004">{{cite book |author-first=John Colin |author-last=Stillwell |author-link=John Colin Stillwell |title=Mathematics and its History |edition=2nd |publisher=[[Springer Science+Business Media Inc.]] |isbn=978-0387953366 |date=2004 |chapter=Infinite Series |page=170 |quote=Infinite series were present in Greek mathematics, [...] There is no question that Zeno's paradox of the dichotomy (Section 4.1), for example, concerns the decomposition of the number 1 into the infinite series <sup>1</sup>⁄<sub>2</sub> + <sup>1</sup>⁄<sub>2</sub><sup>2</sup> + <sup>1</sup>⁄<sub>2</sub><sup>3</sup> + <sup>1</sup>⁄<sub>2</sub><sup>4</sup> + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + <sup>1</sup>⁄<sub>4</sub> + <sup>1</sup>⁄<sub>4</sub><sup>2</sup> + <sup>1</sup>⁄<sub>4</sub><sup>3</sup> + ... = <sup>4</sup>⁄<sub>3</sub>. Both these examples are special cases of the result we express as summation of a geometric series}}</ref>
<ref name="Smith_1958">{{cite book |author-last=Smith |author-first=David Eugene |author-link=David Eugene Smith |date=1958 |title=History of Mathematics |url=https://archive.org/details/historyofmathema0002smit |url-access=registration |publisher=[[Dover Publications]] |isbn=978-0486204307}}</ref>
<ref name="Evans_1998">{{cite book |author-link=Lawrence Craig Evans |author-first=Lawrence Craig |author-last=Evans |title=Partial Differential Equations |publisher=[[American Mathematical Society]] |location=Providence |date=1998 |isbn=978-0821807729}}</ref>
<ref name="Rudin_1991">{{cite book |author-last=Rudin |author-first=Walter |author-link=Walter Rudin |title=Functional Analysis |publisher=[[McGraw-Hill Science]] |date=1991 |isbn=978-0070542365 |url=https://archive.org/details/functionalanalys0000rudi |url-access=registration}}</ref>
<ref name="Conway_1994">{{cite book |author-last=Conway |author-first=John Bligh |author-link=John Bligh Conway |title=A Course in Functional Analysis |edition=2nd |publisher=[[Springer-Verlag]] |date=1994 |isbn=978-0387972459 |url=https://books.google.com/books?id=ix4P1e6AkeIC |access-date=2016-02-11 |archive-date=2020-09-09 |archive-url=https://web.archive.org/web/20200909165657/https://books.google.com/books?id=ix4P1e6AkeIC |url-status=live}}</ref>
<ref name="Ahlfors_1979">{{cite book |author-last=Ahlfors |author-first=Lars Valerian |author-link=Lars Valerian Ahlfors |title=Complex Analysis |location=New York |publisher=[[McGraw-Hill]] |edition=3rd |date=1979 |isbn=978-0070006577 |url=https://books.google.com/books?id=2MRuus-5GGoC }}</ref>
}}

== Further reading ==
* {{anchor|Mathematics: Its Content, Methods, and Meaning}}{{cite book |editor-last1=Aleksandrov |editor-first1=A. D. |editor-link1=Aleksandr Danilovich Aleksandrov |editor-last3=Lavrent'ev |editor-first3=M. A. |editor-link3=Mikhail Alekseevich Lavrentyev |editor-last2=Kolmogorov |editor-first2=A. N. |editor-link2=Andrey Nikolaevich Kolmogorov |translator-first1=S. H. |translator-last1=Gould |volume=1–3 |date=March 1969 |title=Mathematics: Its Content, Methods, and Meaning <!-- |title-link=Mathematics: Its Content, Methods, and Meaning --> |edition=2nd |publisher=[[The M.I.T. Press]] / [[American Mathematical Society]] |publication-place=Cambridge, Massachusetts }}
* {{cite book |author-last=Apostol |author-first=Tom M. |author-link=Tom M. Apostol |date=1974 |title=Mathematical Analysis |edition=2nd |publisher=[[Addison–Wesley]] |isbn=978-0201002881}}
* {{cite book |author-last=Binmore |author-first=Kenneth George |author-link=Kenneth George Binmore |date=1981 |orig-date=1981 |title=The foundations of analysis: a straightforward introduction |url=https://archive.org/details/foundationsofana0000binm |url-access=registration |publisher=[[Cambridge University Press]]}}
* {{cite book |author-last1=Johnsonbaugh |author-first1=Richard |author-link1=Richard Johnsonbaugh |author-first2=William Elmer |author-last2=Pfaffenberger |date=1981 |title=Foundations of mathematical analysis |location=New York |publisher=[[M. Dekker]]}}
* {{cite encyclopedia |author-first=Sergey Mikhailovich [Серге́й Миха́йлович] |author-last=Nikol'skiĭ [Нико́льский] |author-link=Sergey Mikhailovich Nikolsky |date=2002 |url=https://encyclopediaofmath.org/wiki/Mathematical_analysis |title=Mathematical analysis |encyclopedia=[[Encyclopaedia of Mathematics]] |editor-link=Michiel Hazewinkel |editor-first=Michiel |editor-last=Hazewinkel |publisher=[[Springer-Verlag]] |isbn=978-1402006098}}
* {{cite book |author-first1=Nicola |author-last1=Fusco |author-link1=Nicola Fusco |author-first2=Paolo |author-last2=Marcellini |author-link2=Paolo Marcellini |author-first3=Carlo |author-last3=Sbordone |date=1996 |title=Analisi Matematica Due |language=it |publisher={{ill|Liguori Editore|it}} |isbn=978-8820726751}}
* {{cite book |author-last=Rombaldi |author-first=Jean-Étienne |date=2004 |title=Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques |language=fr |publisher=[[EDP Sciences]] |isbn=978-2868836816}}
* {{cite book |title=Principles of Mathematical Analysis |author-last=Rudin |author-first=Walter |author-link=Walter Rudin |publisher=[[McGraw-Hill]] |date=1976 |isbn=978-0070542358 |edition=3rd |location=New York}}
* {{cite book |title=Real and Complex Analysis |author-last=Rudin |author-first=Walter |author-link=Walter Rudin |publisher=[[McGraw-Hill]] |date=1987 |isbn=978-0070542341 |edition=3rd |location=New York}}
* {{cite book |title=A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions |title-link=Whittaker and Watson |author-last1=Whittaker |author-first1=Edmund Taylor |author-link1=Edmund Taylor Whittaker |author-last2=Watson |author-first2=George Neville |author-link2=George Neville Watson |date=1927-01-02 |edition=4th |publisher=[[at the University Press]] |publication-place=Cambridge |isbn=0521067944 }} (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992<!-- 1992? -->)
* {{cite web |url=http://www.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf |archive-url=https://web.archive.org/web/20070419024458/http://www.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf |archive-date=2007-04-19 |url-status=live |title=Real Analysis – Course Notes}}

==External links==
{{Wikiquote}}
{{Commons category}}
* [http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis]
* [http://www.jirka.org/ra/ Basic Analysis: Introduction to Real Analysis] by Jiri Lebl ([[Creative Commons|Creative Commons BY-NC-SA]])
* [https://www.britannica.com/topic/analysis-mathematics Mathematical Analysis – Encyclopædia Britannica]
* [http://mathworld.wolfram.com/topics/CalculusandAnalysis.html Calculus and Analysis]

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[[Category:Mathematical analysis| ]]