تحليل رياضي

A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

يطلق اسم التحليل الرياضي على فرع الرياضيات الذي يهتم بدراسة الدوال الرياضية و تحولاتها باستخدام أدوات ترتبط بمفاهيم النهاية ، حيث تدرس خواص مثل الاستمرار و الاشتقاق و التكامل و التفاضل ، التقعر و الإنعطاف في منحنيات التوابع و الدوال، وغالباً ما تدرس هذه المفاهيم على أعداد حقيقية أو أعداد عقدية والدوال المعرفة عليها ومن الممكن أن تدرس أيضاً على فضاءات أخرى كالفضاء المتري أو الطبولوجي.

التاريخ

أول من عرف باستخدام مفاهيم النهايات والتقارب كان عدد من رياضيي اليونان أمثال اودوكسوس و أرخميدس الذين قاما باستخدام هذه المفاهيم بشكل غير تقليدي عندما استخدما طريقة method of exhaustion لحساب مساحة وحجم المساحات والأجسام. في القرن الثاني عشر قام الرياضي الهندي باسكارا بإعطاء عما يمكن أن ندعوه الان "معامل تفاضلي" وكانت الفكرة الأساسية وراء ما ندعوه حاليا مبرهنة رول. في القرن الرابع عشر قام الرياضياتي الهندي مادهافا من سانغاماغراما بالتعبير عن عدة دوال مثلثية كسلاسل غير متناهية ، قدر مقدار الخطأ في التقديرات التي تعطيها هذه السلاسل .

في أوروبا نشأ التحليل في القرن السابع عشر عن طريق اختراع مستقل لكلا العالمين اسحاق نيوتن وغوتفريد لايبنتز. في القرن السابع عشر والثامن عشر، تطورت تطبيقات مواضيع التحليل مثل حسبان التغيرات والمعادلات التفاضلية النظامية و الجزئية، سلاسل فورييه و الدوال المولدة generating function في الأعمال التطبيقية .كما استخدم التحليل الرياضي لمقاربة مسائل الرياضيات المتقطعة بمثيلاتها المستمرة و نجحت هذه الطريقة في عدة حالات .

خلال القرن الثامن عشر كان تعريف الدالة الرياضي موضع نقاش طيل بين الرياضياتيين . في القرن التاسع عشر، كاوشي كان أول من وضع التحليل على أساس منطقي ثابت بإدخال مفهوم سلسلة كاوشي. كما أنه بدأ بوضع النظرية الشكلية للتحليل المركب (العقدي). سيمون بواسون وليوفيل Liouville و جان-بابتيست جوزيف فورييه وآخرون قاموا بدراسة المعادلات التفاضلية الجزئية والتحليل التوافقي harmonic analysis.

في منتصف القرن قدم بيرنارد ريمان نظريته حول التكامل . جاء بعده كارل فايرشتراس الذي قام بحسبنة arithmetization التحليل في نهاية القرن التاسع عشر ، معبرا عن شكوكه ان البرهنة الهندسية تحوي خللا مضللا و هنا قام بتقديم تعريف ε-δ للنهاية .

بدأ عندها شك الرياضيون بأنهم يفترضون وجود استمرارية continuum في الأعداد الحقيقية بدون برهان . قام عندها ديديكايند بتشكيل الأعداد الحقيقية باستخدام حد ديديكايند Dedekind cut . في ذات الوقت تتالت المحاولات لتحسن مبرهنة تكامل ريمان مما أدى لدراسة "حجم" مجموعة تقطعات discontinuity الدوال الحقيقية .

ضمن هذا السياق ، قام كاميل جوردان بتطوير نظريته حول القياس ، في حين طور كانتور ما يمكن تسميته حاليا بنظرية المجموعات المبسطة ، باير قام بالبرهنة عن مبرهنة تصنيف باير . في أوائل القرن العشرين ، تمت صياغة التحليل الرياضي باستخدام نظرية المجموعات البدهياتية axiomatic set theory. قام هنري لبگ Henri Leon Lebesgue بحل مشكلة القياس ، في حين قام هلبرت بتقديم فضاء هلبرت لحل المعادلات التكاملية . كانت فكرة الفضاء الشعاعي المنظم normed vector space تلوح في الأفق ، في عام 1920 قام ستيفان باناخ بإيجاد التحليل الدالي functional analysis .

المفاهيم الهامة

Metric spaces

المقالة الرئيسية: Metric space

In mathematics, a metric space is a set where a notion of distance (called a 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 spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).

Formally, a metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$, i.e., a function

$d:M\times M\to \mathbb{ℝ}$

such that for any $x,y,z\in M$, the following holds:

1. $d(x,y)\ge 0$, with equality if and only if $x=y$ 
2. $d(x,y)=d(y,x)$    (symmetry), and
3. $d(x,z)\le d(x,y)+d(y,z)$    (triangle inequality).

By taking the third property and letting $z=x$, it can be shown that $d(x,y)\ge 0$     (non-negative).

المتتاليات والنهايات

المقالة الرئيسية: متتالية

انظر أيضاً: نهاية متتالية

المتتالية هي فئة مرتّبة. مثل الفئة، فهي تضم 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 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 (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_n) (with n running from 1 to infinity understood) the distance between a_n and x approaches 0 as n → ∞, denoted

$\underset{n\to \mathrm{\infty }}{\lim }{a}_n=x.$

الدوال المتصلة

المقالة الرئيسية: دالة متصلة

A function $f$ is continuous at a point $x$ in the interior of its domain if for every neighbourhood $V$ of $f(x)$, there exists a neighbourhood $U$ of $x$ such that the image of $U$ under $f$ is a subset of $V$.

Informally, this means that the image of any neighbourhood of $x$ can be made arbitrarily small.

The notion of a continuous function can be generalized to topological spaces. The condition for continuity is a quintessential requirement for many theorems in mathematical analysis.

الفروع الرئيسية

ومن فروع التحليل الرياضي

• تحليل حقيقي.
• تحليل مركب.
• تحليل دالي.
• نظرية القياس.
• معادلات تفاضلية
• معادلات تكاملية
• نظرية التحويلات

الحسبان

المقالة الرئيسية: الحسبان

التحليل الحقيقي

المقالة الرئيسية: Real analysis

Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

Complex analysis

المقالة الرئيسية: Complex analysis

Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers. 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 functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.

Functional analysis

المقالة الرئيسية: 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 operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

Free analysis

المقالة الرئيسية: Noncommutative functional analysis

Free or noncommutative analysis is a sub-branch of functional analysis that deals with spaces that are, in some ways, noncommutative. It is related to noncommutative function theory, which is a noncommutative generalization of complex analysis, as well as free probability theory and noncommutative geometry. In free analysis, emphasis is placed on working with noncommutative variables and functions, especially over spaces or algebras that are noncommutative.

Harmonic analysis

المقالة الرئيسية: Harmonic analysis

Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves. This includes the study of the notions of Fourier series and Fourier transforms (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

المقالة الرئيسية: Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. 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 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 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 equation of motion) may be solved explicitly.

Measure theory

المقالة الرئيسية: Measure (mathematics)

A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. 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 $n$-dimensional Euclidean space ${\mathbb{ℝ}}^n$. For instance, the Lebesgue measure of the interval $[0,1]$ in the 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 +∞ to (certain) subsets of a set $X$. 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. This means that the empty set, countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets 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

المقالة الرئيسية: Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

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 century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Vector analysis

المقالة الرئيسية: Vector calculus

انظر أيضاً: A History of Vector Analysis and Vector Analysis

Vector analysis, also called vector calculus, is a branch of mathematical analysis dealing with vector-valued functions.

Scalar analysis

المقالة الرئيسية: 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

المقالة الرئيسية: Tensor field

مواضيع أخرى

• Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.
• Harmonic analysis deals with the representation of functions or signals as the superposition of basic waves.
• Geometric analysis involves the use of geometrical methods in the study of partial differential equations 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, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
• Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.
• Computable analysis, the study of which parts of analysis can be carried out in a 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 + A = 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, rather than classical, logic and set theory.
• Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences.
• Paraconsistent analysis, which is built upon a foundation of 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, relativity, and quantum mechanics is based on applied analysis, and differential equations 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 audio, radio waves, 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.

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 games
• Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
• Differentiable manifolds
• Differential topology
• Partial differential equations

Notable textbooks

• Leonhard Euler (1748) Introductio in analysin infinitorum
• A. L. Cauchy (1821) Cours d'analyse
• Camille Jordan (1882) Cours_d'analyse de l'École polytechnique
• G. H. Hardy (1908) A Course of Pure Mathematics
• E. T. Whittaker & G. N. Watson (1915) A Course of Modern Analysis
• George Pólya & Gábor Szegő (1925) Problems and Theorems in Analysis (two volumes)
• Walter Rudin (1953) Principles of Mathematical Analysis
• Elias M. Stein & Rami Shakarchi (2003, 2011) Princeton Lectures in Analysis (four volumes)

See also

• Arithmetization of analysis
• Constructive analysis
• History of calculus
• Hypercomplex analysis
• Multiple rule-based problems
• Multivariable calculus
• Paraconsistent logic
• Smooth infinitesimal analysis
• Timeline of calculus and mathematical analysis

Further reading

• Aleksandrov, A. D.; Kolmogorov, A. N.; Lavrent'ev, M. A., eds. (March 1969). Mathematics: Its Content, Methods, and Meaning. Vol. 1–3. Translated by Gould, S. H. (2nd ed.). Cambridge, Massachusetts: The M.I.T. Press / American Mathematical Society.
• Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Addison–Wesley. ISBN 978-0201002881.
• Binmore, Kenneth George (1981) [1981]. The foundations of analysis: a straightforward introduction. Cambridge University Press.
• Johnsonbaugh, Richard; Pfaffenberger, William Elmer (1981). Foundations of mathematical analysis. New York: M. Dekker.
• Nikol'skiĭ [Нико́льский], Sergey Mikhailovich [Серге́й Миха́йлович] (2002). "Mathematical analysis". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Springer-Verlag. ISBN 978-1402006098.
• Fusco, Nicola; Marcellini, Paolo; Sbordone, Carlo (1996). Analisi Matematica Due (in الإيطالية). Liguori Editore. ISBN 978-8820726751.
• Rombaldi, Jean-Étienne (2004). Éléments d'analyse réelle : CAPES et agrégation interne de mathématiques (in الفرنسية). EDP Sciences. ISBN 978-2868836816.
• Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542358.
• Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). New York: McGraw-Hill. ISBN 978-0070542341.
• Whittaker, Edmund Taylor; Watson, George Neville (1927-01-02). 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 (4th ed.). Cambridge: at the University Press. ISBN 0521067944. (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992)
• "Real Analysis – Course Notes" (PDF). Archived (PDF) from the original on 2007-04-19.

وصلات خارجية

• Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
• Basic Analysis: Introduction to Real Analysis by Jiri Lebl (Creative Commons BY-NC-SA)
• Mathematical Analysis – Encyclopædia Britannica
• Calculus and Analysis