Advanced Calculus And Partial Differential Equations
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ADVANCED CALCULUS & PARTIAL DIFFERENTIAL EQUATIONS
–Unit-I– 1.1 Historical background : 1.1.1 A brief historical background of Calculus and partial differential equations in the context of India and Indian heritage and culture 1.1.2 Abrief biography of Bodhayana 1.2 Field structure and ordered structure of R, Intervals, Bounded and Unbounded sets, Supremum and Infimum, Completeness in R, Absolute value of a real number. 1.3 Sequence of real numbers 1.4 Limit of a sequence 1.5 Bounded and Monotonic sequences 1.6 Cauchy’s general principle of convergence 1.7 Algebra of sequence and some important theorems –Unit-II– 2.1 Series of non-negative terms 2.2 Convergence of positive term series 2.3 Alternating series and Leibnitz’s test 2.4 Absolute and Conditional Convergence of Series of real terms 2.5 Uniform continuity 2.6 Chain rule of differentiability 2.7 Mean value theorems and their geometrical interpretations –Unit-III– 3.1 Limit and Continuity of functions of two variables 3.2 Change of variables 3.3 Euler’s theorem on homogeneous functions 3.4 Taylor’s theorem for function of two variables 3.5 Jacobians 3.6 Maxima and Minima of functions of two variables 3.7 Lagrange’s multiplier method 3.8 Beta and Gamma Functions –Unit-IV– 4.1 Partial differential equations of the first order 4.2 Lagrange’s solution 4.3 Some special types of equations which can be solved easily by methods other than the general method 4.4 Charpit’s general method 4.5 Partial differential equations of second and higher orders –Unit-V– 5.1 Classification of partial differential equations of second order 5.2 Homogeneous and non-homogeneous partial differential equations of constant coefficients 5.3 Partial differential equations reducible to equations with constant coefficients
Advanced Calculus (Revised Edition)
Author: Lynn Harold Loomis
language: en
Publisher: World Scientific Publishing Company
Release Date: 2014-02-26
An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.
An Introduction to Partial Differential Equations
Author: Michael Renardy
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-04-18
Partial differential equations are fundamental to the modeling of natural phenomena, arising in every field of science. Consequently, the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis and algebraic topology. Like algebra, topology, and rational mechanics, partial differential equations are a core area of mathematics. This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables. Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses. This new edition features new problems throughout and the problems have been rearranged in each section from simplest to most difficult. New examples have also been added. The material on Sobolev spaces has been rearranged and expanded. A new section on nonlinear variational problems with "Young-measure" solutions appears. The reference section has also been expanded.