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Hyperbolic Sector
A hyperbolic sector is a region (mathematics), region of the Cartesian plane bounded by a hyperbola and two ray (geometry), rays from the origin to it. For example, the two points and on the Hyperbola#Rectangular hyperbola, rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and its orientation (geometry), orientation is altered by a Rotation (geometry), rotation leaving the center at the origin, as with the Unit hyperbola#Parametrization, unit hyperbola. A hyperbolic sector in standard position has and . The argument of hyperbolic functions is the hyperbolic angle, which is defined as the area of a hyperbolic sector of the standard hyperbola ''xy'' = 1. This area is evaluated using natural logarithm. Hyperbolic triangle When in standard position, a hyperbolic sector determines a hyperbolic triangle, the right triangle with one vertex (geometry), vertex at the origin, base on the diagonal ray ''y'' = ''x'', and third vertex on the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Coordinates
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane :\ = Q. Hyperbolic coordinates take values in the hyperbolic plane defined as: :HP = \. These coordinates in ''HP'' are useful for studying logarithmic comparisons of direct proportion in ''Q'' and measuring deviations from direct proportion. For (x,y) in Q take :u = \ln \sqrt and :v = \sqrt. The parameter ''u'' is the hyperbolic angle to (''x, y'') and ''v'' is the geometric mean of ''x'' and ''y''. The inverse mapping is :x = v e^u ,\quad y = v e^. The function Q \rarr HP is a continuous mapping, but not an analytic function. Alternative quadrant metric Since ''HP'' carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence Q \leftrightarrow HP brings this structure to ''Q''. It can be grasped using the notion of hyperbolic motions. Since geodesics in ''HP'' are semicircles with centers on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Humboldt State University
California State Polytechnic University, Humboldt (Cal Poly Humboldt or Humboldt) is a public university in Arcata, California. It is one of Cal Poly (other), three polytechnic universities in the California State University (CSU) system and the northernmost campus in the system. The main campus, situated hillside at the edge of a Sequoia sempervirens, coast redwood forest, overlooks Arcata, much of Humboldt Bay, and the Pacific Ocean. The college town setting on the North Coast, California, California North Coast, north of Eureka, California, Eureka, north of San Francisco, California, San Francisco, and 654 miles (1053 km) north of Los Angeles, is notable for its natural beauty. It is the most westerly four-year university in the contiguous United States. Humboldt is a Hispanic-serving institution (HSI). The university is divided into three colleges: the College of Arts, Humanities, and Social Sciences; the College of Natural Resources and Sciences; and the C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Quadrature Of The Parabola
''Quadrature of the Parabola'' () is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing that the area of a parabolic segment (the region enclosed by a parabola and a line) is \tfrac43 that of a certain inscribed triangle. It is one of the best-known works of Archimedes, in particular for its ingenious use of the method of exhaustion and in the second part of a geometric series. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. This represents the most sophisticated use of a ''reductio ad absurdum'' argument in ancient Greek mathematics, and Archimedes' solution remained unsurpassed until the development of integral calculus in the 17th century, being succee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes
Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenistic Sicily, Sicily. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and mathematical analysis, analysis by applying the concept of the Cavalieri's principle, infinitesimals and the method of exhaustion to derive and rigorously prove many geometry, geometrical theorem, theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Archimedes' other math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cavalieri's Quadrature Formula
In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral :\int_0^a x^n\,dx = \tfrac\, a^ \qquad n \geq 0, and generalizations thereof. This is the definite integral form; the indefinite integral form is: :\int x^n\,dx = \tfrac\, x^ + C \qquad n \neq -1. There are additional forms, listed below. Together with the linearity of the integral, this formula allows one to compute the integrals of all polynomials. The term " quadrature" is a traditional term for area; the integral is geometrically interpreted as the area under the curve ''y'' = ''x''''n''. Traditionally important cases are ''y'' = ''x''2, the quadrature of the parabola, known in antiquity, and ''y'' = 1/''x'', the quadrature of the hyperbola, whose value is a logarithm. Forms Negative ''n'' For negative values of ''n'' (negative powers of ''x''), there is a singularity at ''x'' = 0, and thus the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrature (mathematics)
In mathematics, quadrature is a historic term for the computation of areas and is thus used for computation of integrals. The word is derived from the Latin ''quadratus'' meaning "square". The reason is that, for Ancient Greek mathematicians, the computation of an area consisted of constructing a square of the same area. In this sense, the modern term is squaring. For example, the quadrature of the circle, (or squaring the circle) is a famous old problem that has been shown, in the 19th century, to be impossible with the methods available to the Ancient Greeks, Integral calculus, introduced in the 17th century, is a general method for computation of areas. ''Quadrature'' came to refer to the computation of any integral; such a computation is presently called more often "integral" or "integration". However, the computation of solutions of differential equations and differential systems is also called ''integration'', and ''quadrature'' remains useful for distinguish integra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically as . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called ''differentiation'', which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as and . Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, veloc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Visual Proof Hyperbolic Sector Area
The visual system is the physiological basis of visual perception (the ability to detect and process light). The system detects, transduces and interprets information concerning light within the visible range to construct an image and build a mental model of the surrounding environment. The visual system is associated with the eye and functionally divided into the optical system (including cornea and lens) and the neural system (including the retina and visual cortex). The visual system performs a number of complex tasks based on the ''image forming'' functionality of the eye, including the formation of monocular images, the neural mechanisms underlying stereopsis and assessment of distances to (depth perception) and between objects, motion perception, pattern recognition, accurate motor coordination under visual guidance, and colour vision. Together, these facilitate higher order tasks, such as object identification. The neuropsychological side of visual information proce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Messenger Of Mathematics
The ''Messenger of Mathematics'' is a defunct British mathematics journal. The founding editor-in-chief was William Allen Whitworth with Charles Taylor and volumes 1–58 were published between 1872 and 1929. James Whitbread Lee Glaisher was the editor-in-chief after Whitworth. In the nineteenth century, foreign contributions represented 4.7% of all pages of mathematics in the journal. History The journal was originally titled ''Oxford, Cambridge and Dublin Messenger of Mathematics''. It was supported by mathematics students and governed by a board of editors composed of members of the universities of Oxford, Cambridge and Dublin (the last being its sole constituent college, Trinity College Dublin). Volumes 1–5 were published between 1862 and 1871. It merged with ''The Quarterly Journal of Pure and Applied Mathematics'' to form the ''Quarterly Journal of Mathematics''. References Further reading * External links''Messenger of Mathematics'', vols. 1–30 (1871&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Burnside
:''This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920).'' __NOTOC__ William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early researcher in the theory of finite groups. Burnside was born in London in 1852. He went to school at Christ's Hospital until 1871 and attended St. John's and Pembroke Colleges at the University of Cambridge, where he was the Second Wrangler (bracketed with George Chrystal) in 1875. He lectured at Cambridge for the following ten years, before being appointed professor of mathematics at the Royal Naval College in Greenwich. While this was a little outside the main centres of British mathematical research, Burnside remained a very active researcher, publishing more than 150 papers in his career. Burnside's early research was in applied mathematics. This work was of sufficient distinction to merit his election as a fellow of the Royal Society ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
