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Calculus Of Variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematics), mappings from a set of Function (mathematics), functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and Mathematical notation, notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Kingdom of Prussia, Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase Pi (letter), pi) to denote Pi, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guillaume De L'Hôpital
Guillaume François Antoine, Marquis de l'Hôpital (; sometimes spelled L'Hospital; 7 June 1661 – 2 February 1704) was a French mathematician. His name is firmly associated with l'Hôpital's rule for calculating limits involving indeterminate forms 0/0 and ∞/∞. Although the rule did not originate with l'Hôpital, it appeared in print for the first time in his 1696 treatise on the infinitesimal calculus, entitled '' Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes''. This book was a first systematic exposition of differential calculus. Several editions and translations to other languages were published and it became a model for subsequent treatments of calculus. Biography L'Hôpital was born into a military family. His father was Anne-Alexandre de l'Hôpital, a Lieutenant-General of the King's army, Comte de Saint-Mesme and the first squire of Gaston, Duke of Orléans. His mother was Elisabeth Gobelin, a daughter of Claude Gobelin, Intendant in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Bernoulli
Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibnizian calculus, to which he made numerous contributions. A member of the Bernoulli family, he, along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant . However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work '' Ars Conjectandi''.Jacob (Jacques) Bernoulli [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a polymath. He was born in the city of Pisa, then part of the Duchy of Florence. Galileo has been called the father of observational astronomy, modern-era classical physics, the scientific method, and modern science. Galileo studied speed and velocity, gravity and free fall, the principle of relativity, inertia, projectile motion and also worked in applied science and technology, describing the properties of the pendulum and "hydrostatic balances". He was one of the earliest Renaissance developers of the thermoscope and the inventor of various sector (instrument), military compasses. With an improved telescope he built, he observed the stars of the Milky Way, the phases of Venus, the Galilean moons, four largest satellites of Jupiter, Saturn's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Bernoulli
Johann Bernoulli (also known as Jean in French or John in English; – 1 January 1748) was a Swiss people, Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth. Biography Early life Johann was born in Basel, the son of Nicolaus Bernoulli, an apothecary, and his wife, Margarethe Schongauer, and began studying medicine at University of Basel. His father desired that he study business so that he might take over the family spice trade, but Johann Bernoulli did not like business and convinced his father to allow him to study medicine instead. Johann Bernoulli began studying mathematics on the side with his older brother Jacob Bernoulli. Throughout Johann Bernoulli's education at Basel University, the Bernoulli brothers worked together, spending much of their time studying the newly discovered infinitesimal calculus. They were among t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brachistochrone Curve
In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696 and solved by Isaac Newton in 1697. The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal.Stewart, James. "Section 10.1 - Curves Defined by Parametric Equations." ''Calculus: Early Transcendentals''. 7th ed. Belmont, CA: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World Scientific Publishing
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, with more than 170 journals in various fields. In 1995, World Scientific co-founded the London-based Imperial College Press together with the Imperial College of Science, Technology and Medicine. Company structure The company head office is in Singapore. The Chairman and Editor-in-Chief is Dr Phua Kok Khoo, while the Managing Director is Doreen Liu. The company was co-founded by them in 1981. Imperial College Press In 1995 the company co-founded Imperial College Press, specializing in engineering, medicine and information technology, with Imperial College London Imperial College London, also known as Imperial, is a Public university, public research university in London, England. Its history began with Prince Albert of Saxe-Coburg and Gotha, Prince Albert, h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophiæ Naturalis Principia Mathematica
(English: ''The Mathematical Principles of Natural Philosophy''), often referred to as simply the (), is a book by Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation. The ''Principia'' is written in Latin and comprises three volumes, and was authorized, imprimatur, by Samuel Pepys, then-President of the Royal Society on 5 July 1686 and first published in 1687. The is considered one of the most important works in the history of science. The French mathematical physicist Alexis Clairaut assessed it in 1747: "The famous book of ''Mathematical Principles of Natural Philosophy'' marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton ... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses." The French scientist Joseph-Louis Lagrange described it as "the greatest production of the human mind". French polymath Pierre-Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Minimal Resistance Problem
Newton's minimal resistance problem is a problem of finding a solid of revolution which experiences a minimum resistance when it moves through a homogeneous fluid with constant velocity in the direction of the axis of revolution, named after Isaac Newton, who posed and solved the problem in 1685 and published it in 1687 in his Philosophiæ Naturalis Principia Mathematica, Principia Mathematica. The problem initiated the field of the calculus of variations, as Newton introduced the concept of calculus of variations, with the problem being the first to be formulated and correctly solved, appearing a decade before the Brachistochrone curve, brachistochrone problem, in which Newton also solved using the calculus of variations. Newton published the solution in Philosophiæ Naturalis Principia Mathematica, Principia Mathematica without his derivation, and David Gregory (mathematician), David Gregory was the first person who approached Newton and persuaded him to write an analysis for him. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed. His book (''Mathematical Principles of Natural Philosophy''), first published in 1687, achieved the Unification of theories in physics#Unification of gravity and astronomy, first great unification in physics and established classical mechanics. Newton also made seminal contributions to optics, and Leibniz–Newton calculus controversy, shares credit with German mathematician Gottfried Wilhelm Leibniz for formulating calculus, infinitesimal calculus, though he developed calculus years before Leibniz. Newton contributed to and refined the scientific method, and his work is considered the most influential in bringing forth modern science. In the , Newton formulated the Newton's laws of motion, laws of motion and Newton's law of universal g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
