In mathematics, the term mapping, usually shortened to map, refers to either
There are also a few, less common uses in logic and graph theory.
Mathematical analysis is a branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. 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).
Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In India, the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem.
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. Topically, mathematical logic bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.
Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see logic in computer science for those.
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. The set of values the function may take is termed the range of the function.
For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases). For a function whose domain is a subset of the real numbers, when the function is represented in an xy Cartesian coordinate system, the domain is represented on the x-axis.
In mathematics, an inverse function is a function that reverses another function: if the function f applied to an input x gives a result of y, then applying the inverse function g to y gives the result x, and vice versa. i.e. f(x) = y, and g(y) = x. More directly, g(f(x)) = x, meaning g composed with f form an identity.
A function f that has an inverse is defined as invertible; the inverse function is then uniquely determined by f and is denoted by f −1, read f inverse. Superscripted "−1" does not refer to numerical exponentiation: see composition monoid for explanation of this notation.
In mathematics, a global optimum is a selection from a given domain which provides either the highest value (the global maximum) or lowest value (the global minimum), depending on the objective, when a specific function is applied. For example, for the function
defined on the real numbers, the global maximum occurs at x = 0, where f(x) = 2. For all other values of x, f(x) is smaller.