Ordinary differential equations
Springer, 15 авг. 2006 г. - Всего страниц: 334
There are dozens of books on ODEs, but none with the elegantgeometric insight of Arnol'd's book. Arnol'd puts a clear emphasis on the qualitative andgeometric properties of ODEs and their solutions, ratherthan on theroutine presentation of algorithms for solvingspecial classes of equations.Of course, the reader learnshow to solve equations, but with much more understandingof the systems, the solutions and the techniques. Vector fields and one-parameter groups of transformationscome right from the startand Arnol'd uses this "language"throughout the book. This fundamental difference from thestandard presentation allows him to explain some of the realmathematics of ODEs in a very understandable way and withouthidingthe substance. The text is also rich with examples and connections withmechanics. Where possible, Arnol'd proceeds by physicalreasoning, using it as a convenient shorthand for muchlonger formal mathematical reasoning. This technique helpsthe student get a feel for the subject. Following Arnol'd's guiding geometric and qualitativeprinciples, there are 272 figures in the book, but not asingle complicated formula. Also, the text is peppered withhistoricalremarks, which put the material in context,showing how the ideas have developped since Newton andLeibniz. This book is an excellent text for a course whose goal is amathematical treatment of differential equations and therelated physical systems.
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Proof of Uniqueness
The LotkaVolterra Model
Не показаны другие разделы: 153
algebra amplitude axis basis called characteristic equation chart circle coefficients complex Consider coordinate system Corollary corresponding cycle defined Definition denote derivative diffeomorphism differential equation dilations direction field domain eigenvalues eigenvector energy equation of variations equilibrium position equivalent Euclidean space Euler example exponential extended phase space Find finite first-order formula function graph homogeneous inhomogeneous equation initial condition instant integral curves interval Lemma linear equation linear operator linear transformation manifold mapping matrix motion neighborhood Newton nonzero obtain one-parameter group parameter pendulum perturbation phase curves phase flow phase plane phase space phase velocity phase velocity vector polynomial Problem Proof Prove quasi-homogeneous quasi-polynomials rectifying Remark right-hand side rotation Sect sequence singular points smooth solution of Eq solve space Rn sphere stable subspaces sufficiently small surface tangent theory tion topological variables vector field vector space Wronskian zero