Mathematical analysis

Source: Wikipedia, the free encyclopedia.

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

These theories are usually studied in the context of real and complex numbers and 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 of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

History

Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.

Ancient

Mathematical analysis formally developed in the 17th century during the

arithmetic and geometric series as early as the 4th century BCE.[8]
Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE.[9]

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.[10] In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what is now known as Rolle's theorem.[11]

In the 14th century,

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.

infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems
by continuous ones.

Modernization

In the 18th century,

(ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of 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

decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities
of real functions.

Also, various

measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. 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. Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis
.

Important concepts

Metric spaces

In

metric
) between elements of the set is defined.

Much of analysis happens in some metric space; the most commonly used are the

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 where is a set and is a

metric
on , i.e., a function

such that for any , the following holds:

  1. , with equality if and only if    (identity of indiscernibles),
  2.    (symmetry), and
  3.    (triangle inequality).

By taking the third property and letting , it can be shown that     (non-negative).

Sequences and limits

A sequence is an ordered list. Like a

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 (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denoted

Main branches

Calculus

Real analysis

Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the

smoothness
and related properties of real-valued functions.

Complex analysis

Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates

.

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

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of

integral equations
.

Harmonic analysis

Harmonic analysis is a branch of mathematical analysis concerned with the representation of

tidal analysis, and neuroscience
.

Differential equations

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.[21][22][23] 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

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.[24] 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 -dimensional Euclidean space . For instance, the Lebesgue measure of the interval 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 . It must assign 0 to the

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 -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

Numerical analysis is the study of

symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).[25]

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

Vector analysis is a branch of mathematical analysis dealing with values which have both magnitude and direction. Some examples of vectors include velocity, force, and displacement. Vectors are commonly associated with scalars, values which describe magnitude.[26]

Scalar analysis

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

Other topics

Applications

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

Physical sciences

The vast majority of

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

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.[27]

Other areas of mathematics

Techniques from analysis are used in many areas of mathematics, including:

Famous Textbooks

See also

References

  1. ^ Edwin Hewitt and Karl Stromberg, "Real and Abstract Analysis", Springer-Verlag, 1965
  2. Stillwell, John Colin. "analysis | mathematics". Encyclopædia Britannica. Archived
    from the original on 2015-07-26. Retrieved 2015-07-31.
  3. ^ from the original on 2016-05-17. Retrieved 2015-11-15.
  4. . 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 12 + 122 + 123 + 124 + ... and that Archimedes found the area of the parabolic segment (Section 4.4) essentially by summing the infinite series 1 + 14 + 142 + 143 + ... = 43. Both these examples are special cases of the result we express as summation of a geometric series
  5. .
  6. from the original on 2016-06-11. Retrieved 2015-11-15.
  7. ISBN 978-0-7923-3463-7. Archived from the original on 2016-06-17. Retrieved 2015-11-15., Chapter, p. 279 Archived 2016-05-26 at the Wayback Machine
  8. .
  9. ^ K. B. Basant, Satyananda Panda (2013). "Summation of Convergent Geometric Series and the concept of approachable Sunya" (PDF). Indian Journal of History of Science. 48: 291–313.
  10. from the original on 2019-04-21. Retrieved 2015-11-15.
  11. ^ Rajagopal, C. T.; Rangachari, M. S. (June 1978). "On an untapped source of medieval Keralese Mathematics". Archive for History of Exact Sciences. 18 (2): 89–102.
    S2CID 51861422
    .
  12. ^ Pellegrino, Dana. "Pierre de Fermat". Archived from the original on 2008-10-12. Retrieved 2008-02-24.
  13. ^ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. p. 17.
  14. . 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)
  15. .
  16. .
  17. .
  18. .
  19. from the original on 2020-09-09. Retrieved 2016-02-11.
  20. .
  21. .
  22. from the original on 2019-12-27. Retrieved 2018-10-26.
  23. .
  24. ^ Borisenko, A. I.; Tarapov, I. E. (1979). Vector and Tensor Analysis with Applications (Dover Books on Mathematics). Dover Books on Mathematics.
  25. .
  26. ^ "Introductory Real Analysis". 1970.
  27. ^ "Курс дифференциального и интегрального исчисления. Том I". 1969.
  28. ^ "Основы математического анализа. Том II". 1960.
  29. ^ "Курс дифференциального и интегрального исчисления. Том III". 1960.
  30. ^ The Fundamentals of Mathematical Analysis: International Series in Pure and Applied Mathematics, Volume 1. ASIN 0080134734.
  31. ^ The Fundamentals of Mathematical Analysis: International Series of Monographs in Pure and Applied Mathematics, Vol. 73-II. ASIN 1483213153.
  32. ^ "A Course of Mathematical Analysis Vol 1". 1977.
  33. ^ "A Course of Mathematical Analysis Vol 2". 1987.
  34. ^ Mathematical Analysis I. ASIN 3662569558.
  35. ^ Mathematical Analysis II. ASIN 3662569663.
  36. ^ "A Course of Higher Mathematics Vol 3 1 Linear Algebra". 1964.
  37. ^ "A Course of Higher Mathematics Vol 2 Advanced Calculus". 1964.
  38. ^ "A Course of Higher Mathematics Vol 3-2 Complex Variables Special Functions". 1964.
  39. ^ "A Course of Higher Mathematics Vol 4 Integral and Partial Differential Equations". 1964.
  40. ^ "A Course of Higher Mathematics Vol 5 Integration and Functional Analysis". 1964.
  41. ^ "Differential and Integral Calculus". 1969.
  42. ^ "A Course of Mathematical Analysis". 1960.
  43. ^ Mathematical Analysis: A Special Course. ASIN 1483169561.
  44. ^ "Theory of functions of a real variable (Teoria functsiy veshchestvennoy peremennoy, chapters I to IX)". 1955.
  45. ^ "Theory of functions of a real variable =Teoria functsiy veshchestvennoy peremennoy". 1955.
  46. ^ "Problems in Mathematical Analysis". 1970.
  47. ^ Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions. ASIN 3540636404.
  48. ^ Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry. ASIN 3540636862.
  49. ^ Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd Edition. ASIN 0201002884.
  50. ^ Principles of Mathematical Analysis. ASIN 0070856133.
  51. ^ Real Analysis: Measure Theory, Integration, and Hilbert Spaces. ASIN 0691113866.
  52. .
  53. ^ Complex Analysis. ASIN 0691113858.
  54. ^ Functional Analysis: Introduction to Further Topics in Analysis. ASIN 0691113874.
  55. ^ Analysis I: Third Edition. ASIN 9380250649.
  56. ^ Analysis II: Third Edition. ASIN 9380250657.
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  58. .
  59. .
  60. .
  61. .

Further reading

External links