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High Quality Content by WIKIPEDIA articles! In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis) that lies on the same plane. Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid, times the figure's area (Pappus's second centroid Theorem). Rotating a curve A representative disk is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length w) around some axis (located r units away), so that a cylindrical volume of ??r2w units is enclosed.

High Quality Content by WIKIPEDIA articles! Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain partial differential equations (PDEs), often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential convergence" being the fastest possible. PDEs describe a wide array of physical processes such as heat conduction, fluid flow, and sound propagation. In many such equations, there are underlying "basic waves" that can be used to give efficient algorithms for computing solutions to these PDEs. In a typical case, spectral methods take advantage of this fact by writing the solution as its Fourier series, substituting this series into the PDE to get a system of ordinary differential equations (ODEs) in the time-dependent coefficients of the trigonometric terms in the series (written in complex exponential form), and using a...

Ella Hawley

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Xu Jiagu

Olympiad mathematics is not a collection of techniques of solving mathematical problems but a system for advancing mathematical education. This book is based on the lecture notes of the mathematical Olympiad training courses conducted by the author in Singapore. Its scope and depth not only covers and exceeds the usual syllabus, but introduces a variety concepts and methods in modern mathematics. In each lecture, the concepts, theories and methods are taken as the core. The examples are served to explain and enrich their intension and to indicate their applications. Besides, appropriate number of test questions is available for reader's practice and testing purpose. Their detailed solutions are also conveniently provided. The examples are not very complicated so that readers can easily understand. There are many real competition questions included which students can use to verify their abilities. These test questions are from many countries, e.g. China, Russia, USA, Singapore, etc. In...

Krishan L. Duggal, Bayram Sahin

This is the first systematic account of the main results in the theory of lightlike submanifolds of semi-Riemannian manifolds which have a geometric structure, such as almost Hermitian, almost contact metric or quaternion Kahler. Using these structures, the book presents interesting classes of submanifolds whose geometry is very rich. The book also includes hypersurfaces of semi-Riemannian manifolds, their use in general relativity and Osserman geometry, half-lightlike submanifolds of semi-Riemannian manifolds, lightlike submersions, screen conformal submersions, and their applications in harmonic maps. Basic constructions and definitions are presented as preliminary background in every chapter. The presentation explores applications and suggests several open questions. This self-contained monograph provides up-to-date research in lightlike geometry and is intended for graduate students and researchers just entering this field.

William L. Briggs, Lyle Cochran

Drawing on their decades of teaching experience, William Briggs and Lyle Cochran have created a calculus text that carries the teacher’s voice beyond the classroom. That voice–evident in the narrative, the figures, and the questions interspersed in the narrative–is a master teacher leading readers to deeper levels of understanding. The authors appeal to readers’ geometric intuition to introduce fundamental concepts and lay the foundation for the more rigorous development that follows. Comprehensive exercise sets have received praise for their creativity, quality, and scope. Functions; Limits; Derivatives; Applications of the Derivative; Integration; Applications of Integration; Integration Techniques; Sequences and Infinite Series; Power Series; Parametric and Polar Curves. For all readers interested in single variable and multivariable calculus for mathematics, engineering, and science.

High Quality Content by WIKIPEDIA articles! In mathematics, the Seifert?van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.

High Quality Content by WIKIPEDIA articles! In probability theory, the probability space, or probability triple, is a concept which serves as a rigorous mathematical ground for the conventional idea of randomness. It is a mathematical model of a real-world situation (or ?experiment?) where we recognize that certain things occur ?at random?. The model works as following: first, at the outset of the experiment we attempt to envision all possible outcomes which might possibly happen, the set of all such outcomes is called the sample space ?. Second, we recognize that the elementary outcomes could be too little of practical use, and that the more complicated events, consisting possibly of many different elementary outcomes, are of more interest. The collection of all such events is called the ?-algebra scriptstyle mathcal{F}. Third, we have to specify how likely one or another event is going to happen, which is done using the probability measure function P. These three components...

High Quality Content by WIKIPEDIA articles! A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters.If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, like a rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, like a lentil. If the generating ellipse is a circle, the result is a sphere.

High Quality Content by WIKIPEDIA articles! In abstract algebra, the split-quaternions or coquaternions are elements of an associative algebra introduced by James Cockle in 1849 under the latter name. They are also known as para-quaternions (particularly in recent literature on para-quaternionic geometry) or hyperbolic quaternions, although historically the latter term has a different meaning. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain zero divisors, nilpotent elements, and nontrivial idempotents.