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Integral Sign
The integral symbol (''see below'') is used to denote integrals and antiderivatives in mathematics, especially in calculus. History The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings; it first appeared publicly in the article "" (On a hidden geometry and analysis of indivisibles and infinites), published in ''Acta Eruditorum'' in June 1686. The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands. Typography in Unicode and LaTeX Fundamental symbol The integral symbol is in Unicode and \int in LaTeX. In HTML, it is written as ∫ (hexadecimal), ∫ (decimal) and ∫ ( named entity). The original IBM PC code page 437 character set included a couple of characters ⌠,⎮ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IBM PC
The IBM Personal Computer (model 5150, commonly known as the IBM PC) is the first microcomputer released in the List of IBM Personal Computer models, IBM PC model line and the basis for the IBM PC compatible ''de facto'' standard. Released on August 12, 1981, it was created by a team of engineers and designers at IBM, International Business Machines (IBM), directed by William C. Lowe and Philip Don Estridge in Boca Raton, Florida. Powered by an x86-architecture Intel 8088 processor, the machine was based on open architecture and third-party peripherals. Over time, expansion cards and software technology increased to support it. The PC had influence of the IBM PC on the personal computer market, a substantial influence on the personal computer market; the specifications of the IBM PC became one of the most popular computer design standards in the world. The only significant competition it faced from a non-compatible platform throughout the 1980s was from Apple Inc., Apple's Maci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Integral
In 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 over this surface a scalar field (that is, a function of position which returns a scalar as a value), 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. Surface integrals have applications in physics, particularly in the classical theories of electromagnetism and fluid mechanics. Surface integrals of scalar fields Assume that ''f'' is a scalar, vector, or tensor field defined on a surface ''S''. To find an explicit formula for the surface integral of ''f'' over ''S'', we need to parameterize ''S'' by defining a system of curvilinear coordinates on ''S'', like the latitude and longitude on a sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only Connected space, connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP''n'' is a closed ''n''-dimensional manifold. The complex projective space CP''n'' is a closed 2''n''-dimensional manifold. A Real line, line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If M is a closed connected n-manifold, the n-th homology group H_(M;\mathbb) is \mathbb or 0 depending on whether M is Orientability, orientable or not. Moreover, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterclockwise
Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to the left, and back up to the top. The opposite sense of rotation or revolution is (in Commonwealth English) anticlockwise (ACW) or (in North American English) counterclockwise (CCW). Three-dimensional rotation can have similarly defined senses when considering the corresponding angular velocity vector. Terminology Before clocks were commonplace, the terms " sunwise" and "deasil", "deiseil" and even "deocil" from the Scottish Gaelic language and from the same root as the Latin "dexter" ("right") were used for clockwise. " Widdershins" or "withershins" (from Middle Low German "weddersinnes", "opposite course") was used for counterclockwise. The terms clockwise and counterclockwise can only be applied to a rotational motion once a side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contour Integral
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. It also has various applications in physics. Contour integration methods include: * direct integration of a complex-valued function along a curve in the complex plane * application of the Cauchy integral formula * application of the residue theorem One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums. Curves in the complex plane In complex analysis, a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadruple Integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in \mathbb^3 (real-number 3D space) are called triple integrals. For repeated antidifferentiation of a single-variable function, see the Cauchy formula for repeated integration. Introduction Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the -axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Esh (letter)
Esh (majuscule: Ʃ, minuscule: ʃ) is a character used in phonology to represent the voiceless postalveolar fricative (English , as in ''"ship"''). In Unicode, these letters are encoded as and Form, usage, and history Its lowercase form is similar to an integral sign or a long s with an extra leftward hook at the bottom; in 1928 the Africa Alphabet borrowed the Greek letter sigma for the uppercase form . The lowercase form was introduced by Isaac Pitman in his 1847 Phonotypic Alphabet to represent the voiceless postalveolar fricative (English ''sh''). It is not common around African languages, but it is, in fact, used in some, for example, Serer uses its lowercase form to make the [] consonant. This letter is used as well as in the International Phonetic Alphabet. The International Phonetic Alphabet (IPA) uses to represent a Voiceless postalveolar fricative, voiceless palato-alveolar sibilant. Related Obsolete and nonstandard symbols in the International Phoneti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miscellaneous Technical
Miscellaneous Technical is a Unicode block ranging from U+2300 to U+23FF. It contains various common symbols which are related to and used in the various technical, programming language, and academic professions. For example: * Symbol ⌂ (HTML hexadecimal code is ⌂) represents a house or a home. * Symbol ⌘ (⌘) is a "place of interest" sign. It may be used to represent the '' Command key'' on a Mac keyboard. * Symbol ⌚ (⌚) is a watch (or clock). * Symbol ⏏ (⏏) is the "Eject" button symbol found on electronic equipment. * Symbol ⏚ (⏚) is the " Earth Ground" symbol found on electrical or electronic manual, tag and equipment. It also includes most of the uncommon symbols used by the APL programming language. Miscellaneous Technical (2300–23FF) in Unicode In Unicode, ''Miscellaneous Technical'' symbols placed in the hexadecimal range 0x2300–0x23FF, (decimal 8960–9215), as described below. (2300–233F) :1. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
