Multiple integralIn mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
Injective functionIn mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, x1 ≠ x2 implies f(x1) f(x2). (Equivalently, f(x1) = f(x2) implies x1 = x2 in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.
Volume integralIn mathematics (particularly multivariable calculus), a volume integral (∫∫∫) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding density function.
Spherical coordinate systemIn mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin; its polar angle measured from a fixed polar axis or zenith direction; and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the fixed axis, measured from another fixed reference direction on that plane.
Inverse functionIn mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by For a function , its inverse admits an explicit description: it sends each element to the unique element such that f(x) = y. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by 5 then subtracts 7 from the result.
Cylindrical coordinate systemA cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane containing the purple section). The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.
Partial functionIn mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function. More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation.
Surface integralIn mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.
Nowhere continuous functionIn mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some such that for every we can find a point such that and . Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
Surjective functionIn mathematics, a surjective function (also known as surjection, or onto function ˈɒn.tuː) is a function f such that every element y can be mapped from some element x such that f(x) = y. In other words, every element of the function's codomain is the of one element of its domain. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.
Coordinate systemIn geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring.
Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
Uniform continuityIn mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number , then there is a positive real number such that at any and in any function interval of the size .
Gaussian integralThe Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution.
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.
Del in cylindrical and spherical coordinatesThis is a list of some vector calculus formulae for working with common curvilinear coordinate systems. This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): The polar angle is denoted by : it is the angle between the z-axis and the radial vector connecting the origin to the point in question. The azimuthal angle is denoted by : it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
Polar coordinate systemIn mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth.
Continuous linear operatorIn functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.
Integral equationIn mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: where is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.
AreaArea is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.
