Function of several real variablesIn mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex.
Function of a real variableIn mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
Limit of a functionAlthough the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.
Differentiable functionIn mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative exists.
Implicit functionIn mathematics, an implicit equation is a relation of the form where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. For example, the equation of the unit circle defines y as an implicit function of x if −1 ≤ x ≤ 1, and y is restricted to nonnegative values.
Level setIn mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is: When the number of independent variables is two, a level set is called a level curve, also known as contour line or isoline; so a level curve is the set of all real-valued solutions of an equation in two variables x_1 and x_2. When n = 3, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables x_1, x_2 and x_3.
Graph of a functionIn mathematics, the graph of a function is the set of ordered pairs , where In the common case where and are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane. In the case of functions of two variables, that is functions whose domain consists of pairs the graph usually refers to the set of ordered triples where instead of the pairs as in the definition above. This set is a subset of three-dimensional space; for a continuous real-valued function of two real variables, it is a surface.
Function of several complex variablesThe theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi.
Multivariable calculusMultivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one. Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus.
Real-valued functionIn mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions. Let be the set of all functions from a set X to real numbers .
Convex functionIn mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.
Analytic functionIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about converges to the function in some neighborhood for every in its domain.
One-sided limitIn calculus, a one-sided limit refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right. The limit as decreases in value approaching ( approaches "from the right" or "from above") can be denoted: The limit as increases in value approaching ( approaches "from the left" or "from below") can be denoted: If the limit of as approaches exists then the limits from the left and from the right both exist and are equal.
Limit of a sequenceAs the positive integer becomes larger and larger, the value becomes arbitrarily close to . We say that "the limit of the sequence equals ." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
Limit (category theory)In , a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as , and inverse limits. The of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, s and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
Limit inferior and limit superiorIn mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Parental leaveParental leave, or family leave, is an employee benefit available in almost all countries. The term "parental leave" may include maternity, paternity, and adoption leave; or may be used distinctively from "maternity leave" and "paternity leave" to describe separate family leave available to either parent to care for small children. In some countries and jurisdictions, "family leave" also includes leave provided to care for ill family members. Often, the minimum benefits and eligibility requirements are stipulated by law.
Inverse limitIn mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any although their existence depends on the category that is considered. They are a special case of the concept of in category theory. By working in the , that is by reverting the arrows, an inverse limit becomes a direct limit or inductive limit, and a limit becomes a colimit.
X-rayX-ray radiation, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 nanometers to 10 picometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz (3e16Hz to 3e19Hz) and energies in the range 124 keV to 145 eV, respectively. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays.
X-ray tubeAn X-ray tube is a vacuum tube that converts electrical input power into X-rays. The availability of this controllable source of X-rays created the field of radiography, the imaging of partly opaque objects with penetrating radiation. In contrast to other sources of ionizing radiation, X-rays are only produced as long as the X-ray tube is energized. X-ray tubes are also used in CT scanners, airport luggage scanners, X-ray crystallography, material and structure analysis, and for industrial inspection.
