Introduce yourself.

Familiarise yourself.

Rules to follow.

Concern the study of the rules of operations and relations, and the constructions and concepts arising from them.

Used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations.

Deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers.

Examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms).

Studies the behavior of dynamical systems that are highly sensitive to initial conditions; an effect which is popularly referred to as the butterfly effect.

Include counting the structures of a given kind and size, deciding when certain criteria can be met and constructing and analyzing objects meeting the criteria

Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions).

Deals with the behavior of dynamical systems. The desired output of a system is called the reference. A controller manipulates the inputs to obtain the desired effect on the output.

Measure of how a function changes as its input changes. Derivative can be thought of as how much one quantity is changing in response to changes in some other quantity.

Concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.

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.

Discipline using the techniques of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry.

The process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step for numerical evaluation.

A concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space.

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.

Any equation that specifies a function in implicit form. Often, the equation relates the value of a function (or functions) at some point with its values at other points.

Models strategic situations, or games, in which an individual's success in making choices depends on the choices of others.

Concern with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.

Structures used to model pairwise relations between objects from a certain collection. A graph can be to a collection of vertices or 'nodes' or describe a function.

Studies the algebraic structures known as groups. The concept of a group is central to abstract algebra. Algebraic structures can be seen as groups endowed with additional operations and axioms.

A branch of applied mathematics and electrical engineering involving the quantification of information. based on probability theory and statistics.

Integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known.

In constructive logic, a statement is only true if there is a constructive proof that it is true, and only false if there is a constructive proof that it is false.

Used to describe the value that a function or sequence "approaches" as the input or index approaches some value.

The logarithm of a number to a given base is the exponent by which the base must be raised to produce that number.

A measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. A generalization of length, area, volume.

Concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.

Is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one.

Concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.

Study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis

Refers to the selection of a best element from some set of available alternatives. This means solving problems in which one seeks to minimize or maximize a real function.

Studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another.

An expression of finite length constructed from variables and constants, using operations of addition, subtraction, multiplication, and non-negative integer exponents.

A power law is a special kind of mathematical relationship between two quantities. The frequency is said to follow a power law.

Concern analysis of random phenomena. Probability theory is essential to many human activities that involve quantitative analysis of large sets of data.

Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system.

Studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

An inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory to a variety of mathematical spaces.