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Lists of mathematics topics

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Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference tables. They also cover equations named after people, societies, mathematicians, journals, and meta-lists.

The purpose of this list is not similar to that of the Mathematics Subject Classification formulated by the American Mathematical Society. Many mathematics journals ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. This list has some items that would not fit in such a classification, such as list of exponential topics and list of factorial and binomial topics, which may surprise the reader with the diversity of their coverage.

Basic mathematics

This branch is typically taught in secondary education or in the first year of university.

Areas of advanced mathematics

See also: Areas of mathematics and Glossary of areas of mathematics

As a rough guide, this list is divided into pure and applied sections although in reality, these branches are overlapping and intertwined.

Core mathematics

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Algebra

  • Quadratic equations and functions
  • Polynomials
  • Equations and inequalities
  • Sequences and series
  • Exponential and logarithmic functions
  • Binomial theorem
  • Matrices
  • Proof by induction
  • Complex numbers
  • Partial fractions
  • Vectors
  • Trigonometry

Calculus

  • Differentiation (including chain rule and implicit differentiation)
  • Integration (including integration by substitution and by parts)
  • Applications of differentiation (e.g., rates of change, optimization)
  • Applications of integration (e.g., area under curves, volumes of revolution)
  • Differential equations (basic concepts)

Trigonometry:

  • Trigonometric ratios and identities
  • Trigonometric equations
  • Trigonometric functions and their graphs
  • Solutions of triangles

Coordinate Geometry:

  • Cartesian coordinates
  • Straight lines
  • Circles
  • Parabolas, ellipses, and hyperbolas

Sequences and Series:

  • Arithmetic and geometric sequences
  • Arithmetic and geometric series
  • Sigma notation

Exponentials and Logarithms:

  • Properties of exponents and logarithms
  • Solving exponential and logarithmic equations

Proof:

  • Various proof techniques and strategies

Differentiation and Integration Techniques:

  • Trigonometric functions differentiation and integration
  • Exponential and logarithmic functions differentiation and integration

Numerical Methods:

  • Numerical approximation of roots and areas under curves

Applied mathematics

Dynamical systems and differential equations

Phase portrait of a continuous-time dynamical system, the Van der Pol oscillator.

A differential equation is an equation involving an unknown function and its derivatives.

In a dynamical system, a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.

Mathematical physics

Mathematical physics is concerned with "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories".1

Theory of computation

Ray tracing is a process based on computational mathematics.

The fields of mathematics and computing intersect both in computer science, the study of algorithms and data structures, and in scientific computing, the study of algorithmic methods for solving problems in mathematics, science, and engineering.

Information theory and signal processing

Information theory is a branch of applied mathematics and Social science involving the quantification of information. Historically, information theory was developed to find fundamental limits on compressing and reliably communicating data.

Signal processing is the analysis, interpretation, and manipulation of signals. Signals of interest include sound, images, biological signals such as ECG, radar signals, and many others. Processing of such signals includes filtering, storage and reconstruction, separation of information from noise, compression, and feature extraction.

Probability and statistics

The "bell curve"—the probability density function of the normal distribution.
Main article: Lists of statistics topics

Probability theory is the formalization and study of the mathematics of uncertain events or knowledge. The related field of mathematical statistics develops statistical theory with mathematics. Statistics, the science concerned with collecting and analyzing data, is an autonomous discipline (and not a subdiscipline of applied mathematics).

Game theory

Game theory is a branch of mathematics that uses models to study interactions with formalized incentive structures ("games"). It has applications in a variety of fields, including economics, anthropology , political science, social psychology and military strategy.

Operations research

Operations research is the study and use of mathematical models, statistics, and algorithms to aid in decision-making, typically with the goal of improving or optimizing the performance of real-world systems.

Methodology

Mathematical statements

A mathematical statement amounts to a proposition or assertion of some mathematical fact, formula, or construction. Such statements include axioms and the theorems that may be proved from them, conjectures that may be unproven or even unprovable, and also algorithms for computing the answers to questions that can be expressed mathematically.

General concepts

Mathematical objects

Among mathematical objects are numbers, functions, sets, a great variety of things called "spaces" of one kind or another, algebraic structures such as rings, groups, or fields, and many other things.

Equations named after people

About mathematics

Mathematicians

Main article: List of mathematicians

Mathematicians study and research in all the different areas of mathematics. The publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics).

Work of particular mathematicians

See also: Category:Lists of things named after mathematicians

Reference tables

Integrals

In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. The following pages list the integrals of many different functions.

Journals

Meta-lists

See also

Others

Notes

  • ^Note 1 : Definition from the Journal of Mathematical Physics [1].
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