Mathematical analysis

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

Branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

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

57 related topics

Alpha

An overview of differences in spelling across English dialects.

Rigour

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness.

An overview of differences in spelling across English dialects.

During the 19th century, the term "rigorous" began to be used to describe increasing levels of abstraction when dealing with calculus which eventually became known as mathematical analysis.

Bernard Bolzano

Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views.

Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views.

To this end, he was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik (1810), Der binomische Lehrsatz (1816) and Rein analytischer Beweis (1817).

Representation theory studies how algebraic structures "act" on objects. A simple example is how the symmetries of regular polygons, consisting of reflections and rotations, transform the polygon.

Representation theory

Branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

Branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

Representation theory studies how algebraic structures "act" on objects. A simple example is how the symmetries of regular polygons, consisting of reflections and rotations, transform the polygon.

Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Limit (mathematics)

Value that a function approaches as the input (or index) approaches some value.

Value that a function approaches as the input (or index) approaches some value.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

A proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.

Generality of algebra

Phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange, particularly in manipulating infinite series.

Phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange, particularly in manipulating infinite series.

A proof from Euclid's Elements (c. 300 BC), widely considered the most influential textbook of all time.

In works such as Cours d'Analyse, Cauchy rejected the use of "generality of algebra" methods and sought a more rigorous foundation for mathematical analysis.

Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.

Applied mathematics

Application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry.

Application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry.

Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.
A numerical solution to the heat equation on a pump casing model using the finite element method.
Fluid mechanics is often considered a branch of applied mathematics and mechanical engineering.
Mathematical finance is concerned with the modelling of financial markets.
The Brown University Division of Applied Mathematics is the oldest applied math program in the U.S.
Applied mathematics has substantial overlap with statistics.

Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability.

George Pólya, circa 1973

George Pólya

Hungarian mathematician.

Hungarian mathematician.

George Pólya, circa 1973

He remained a Professor Emeritus at Stanford for the rest of his career, working on a range of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability.

Gottfried Wilhelm Leibniz argued that idealized numbers containing infinitesimals be introduced.

Nonstandard analysis

Fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

Fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

Gottfried Wilhelm Leibniz argued that idealized numbers containing infinitesimals be introduced.

More generally, nonstandard analysis is any form of mathematics that relies on nonstandard models and the transfer principle.

Portrait after Frans Hals

René Descartes

French philosopher, mathematician, scientist and lay Catholic who invented analytic geometry, linking the previously separate fields of geometry and algebra.

French philosopher, mathematician, scientist and lay Catholic who invented analytic geometry, linking the previously separate fields of geometry and algebra.

Portrait after Frans Hals
The house where Descartes was born in La Haye en Touraine
Graduation registry for Descartes at the University of Poitiers, 1616
In Amsterdam, Descartes lived at Westermarkt 6 (Maison Descartes, left).
René Descartes at work
L'homme (1664)
Cover of Meditations
A Cartesian coordinates graph, using his invented x and y axes
Handwritten letter by Descartes, December 1638
Principia philosophiae, 1644

He is credited as the father of analytic geometry, the bridge between algebra and geometry—used in the discovery of infinitesimal calculus and analysis.

The dichotomy

Zeno's paradoxes

Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c.

Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c.

The dichotomy
Achilles and the tortoise
The arrow
The moving rows

Today's analysis achieves the same result, using limits (see convergent series).