|
Monge Equation
In the mathematical theory of partial differential equations, a Monge equation, named after Gaspard Monge, is a first-order partial differential equation In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of ''n'' variables. The equation takes the form : F(x_1,\ldots,x_n,u,u_,\ldots u_) =0. \, S ... for an unknown function ''u'' in the independent variables ''x''1,...,''x''''n'' :F\left(u,x_1,x_2,\dots,x_n,\frac,\dots,\frac\right)=0 that is a polynomial in the partial derivatives of ''u''. Any Monge equation has a Monge cone. Classically, putting ''u'' = ''x''0, a Monge equation of degree ''k'' is written in the form :\sum_ P_(x_0,x_1,\dots,x_k) \, dx_0^ \, dx_1^ \cdots dx_n^=0 and expresses a relation between the differentials ''dx''''k''. The Monge cone at a given point (''x''0, ..., ''x''''n'') is the zero locus of the equation in the tangent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaspard Monge
Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During the French Revolution he served as the Minister of the Marine, and was involved in the reform of the French educational system, helping to found the École Polytechnique. Biography Early life Monge was born at Beaune, Côte-d'Or, the son of a merchant. He was educated at the college of the Oratorians at Beaune. In 1762 he went to the Collège de la Trinité at Lyon, where, one year after he had begun studying, he was made a teacher of physics at the age of just seventeen. After finishing his education in 1764 he returned to Beaune, where he made a large-scale plan of the town, inventing the methods of observation and constructing the necessary instruments; the plan was presented to the town, and is still preserved in their library. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Partial Differential Equation
In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of ''n'' variables. The equation takes the form : F(x_1,\ldots,x_n,u,u_,\ldots u_) =0. \, Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. If a family of solutions of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations. General solution and complete integral The ''general solution'' to the first order partial differential equation is a solution which contains an arbitrary function. But, the sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monge Cone
In the mathematical theory of partial differential equations (PDE), the Monge cone is a geometrical object associated with a first-order equation. It is named for Gaspard Monge. In two dimensions, let :F(x,y,u,u_x,u_y) = 0\qquad\qquad (1) be a PDE for an unknown real-valued function ''u'' in two variables ''x'' and ''y''. Assume that this PDE is non-degenerate in the sense that F_ and F_ are not both zero in the domain of definition. Fix a point (''x''0, ''y''0, ''z''0) and consider solution functions ''u'' which have :z_0 = u(x_0, y_0).\qquad\qquad (2) Each solution to (1) satisfying (2) determines the tangent plane to the graph :z = u(x,y)\, through the point x_0,y_0,z_0. As the pair (''u''''x'', ''u''''y'') solving (1) varies, the tangent planes envelope a cone in R3 with vertex at x_0, y_0, z_0, called the Monge cone. When ''F'' is quasilinear, the Monge cone degenerates to a single line called the Monge axis. Otherwise, the Monge cone is a proper cone since a nontrivial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Of A Function
In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by :dy = f'(x)\,dx, where f'(x) is the derivative of ''f'' with respect to ''x'', and ''dx'' is an additional real variable (so that ''dy'' is a function of ''x'' and ''dx''). The notation is such that the equation :dy = \frac\, dx holds, where the derivative is represented in the Leibniz notation ''dy''/''dx'', and this is consistent with regarding the derivative as the quotient of the differentials. One also writes :df(x) = f'(x)\,dx. The precise meaning of the variables ''dy'' and ''dx'' depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monge–Ampère Equation
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of ''u'' and in the second-order partial derivatives of ''u''. The independent variables (''x'',''y'') vary over a given domain ''D'' of R2. The term also applies to analogous equations with ''n'' independent variables. The most complete results so far have been obtained when the equation is elliptic. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 and later by André-Marie Ampère in 1820. Important results in the theory of Monge–Ampère equations have been obtained by Sergei Bernstein, Aleksei Pogorelov, Charles Fefferman, and Louis Nir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encyclopedia Of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, and the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains animations and three-dimensional objects. The encyclopedia has been translated from the Soviet ''Matematicheskaya entsiklopediya'' (1977) originally edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles. Until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. ''Encyclopedia of Mathematics'' wiki A new dynamic version of the encyclopedia is now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kluwer Academic Publishers
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
