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.
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.
Line integralIn mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).
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.
IntegralIn mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.
Curvilinear coordinatesIn geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.
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.
Leibniz integral ruleIn calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form where and the integrands are functions dependent on the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative. It is named after Gottfried Leibniz.
Surface of revolutionA surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) one full revolution around an axis of rotation (normally not intersecting the generatrix, except at its endpoints). The volume bounded by the surface created by this revolution is the solid of revolution. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis.
Socialist calculation debateThe socialist calculation debate, sometimes known as the economic calculation debate, was a discourse on the subject of how a socialist economy would perform economic calculation given the absence of the law of value, money, financial prices for capital goods and private ownership of the means of production. More specifically, the debate was centered on the application of economic planning for the allocation of the means of production as a substitute for capital markets and whether or not such an arrangement would be superior to capitalism in terms of efficiency and productivity.
Calculation in kindNOTOC Calculation in kind or calculation in-natura is a way of valuating resources and a system of accounting that uses disaggregated physical magnitudes as opposed to a common unit of calculation. As the basis for a socialist economy, it was proposed to replace money and financial calculation. In an in-kind economy products are produced for their use values (their utility) and accounted in physical terms. By contrast, in money-based economies, commodities are produced for their exchange value and accounted in monetary terms.
CylinderA cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces.
Economic calculation problemThe economic calculation problem (sometimes abbreviated ECP) is a criticism of using economic planning as a substitute for market-based allocation of the factors of production. It was first proposed by Ludwig von Mises in his 1920 article "Economic Calculation in the Socialist Commonwealth" and later expanded upon by Friedrich Hayek. In his first article, Mises described the nature of the price system under capitalism and described how individual subjective values (while criticizing other theories of value) are translated into the objective information necessary for rational allocation of resources in society.
Yo (Cyrillic)Yo, Jo or Io (Ё ё; italics: Ё ё; jɵ) is a letter of the Cyrillic script. In Unicode, the letter is named CYRILLIC CAPITAL/SMALL LETTER IO. In English, the letter Yo is romanized using the Latin ë (according to the ALA–LC and British Standards), ë (yë word-initially) (BGN/PCGN) or yo/jo (orthographic transcription) for Russian, and as i͡o (ALA–LC), yo (BGN/PCGN), or ë (BSI) for Belarusian. In international systems, Yo is romanized as ë (ISO 9). It was derived from the Cyrillic letter Ye (Е е).
Parametric surfaceA parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.
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.
Improper integralIn mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not continuous functions.
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.
Elliptic integralIn integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.
High-definition televisionHigh-definition television (HD or HDTV) describes a television system which provides a substantially higher than the previous generation of technologies. The term has been used since 1936; in more recent times, it refers to the generation following standard-definition television (SDTV), often abbreviated to HDTV or HD-TV. It is the current de facto standard video format used in most broadcasts: terrestrial broadcast television, cable television, satellite television and Blu-ray Discs.
