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Tangent Half-angle Substitution
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of x into an ordinary rational function of t by setting t = \tan \tfrac x2. This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation formula is: \int f(\sin x, \cos x)\, dx =\int f \frac. The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. Leonhard Euler used it to evaluate the integral \int dx / (a + b\cos x) in his 1768 integral calculus textbook, and Adrien-Marie Legendre described the general method in 1817. The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. It is known in Russia as the universal trigonometric substitution, and also known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Spivak
Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. Spivak was the author of the five-volume ''A Comprehensive Introduction to Differential Geometry''. Biography Spivak was born in Queens, New York (state), New York. He received an Bachelor of Arts, A.B. from Harvard University in 1960, while in 1964 he received a Doctor of Philosophy, Ph.D. from Princeton University under the supervision of John Milnor, with thesis ''On Spaces Satisfying Poincaré Duality''. In 1985 Spivak received the Leroy P. Steele Prize. Spivak lectured on elementary physics. Spivak's book, ''Physics for Mathematicians: Mechanics I'' (published December 6, 2010), contains the material that these lectures stemmed from and more. Spivak was also the designer of the MathTime Professional 2 fonts (which are wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
