Mathematics Advanced Calculus In Hindi
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MATHEMATICS-ADVANCED CALCULUS-IN HINDI
Unit-1 Definition of a sequence. Theorems on limits of sequences. Indeterminate Forms [Appendix (Chapter 19)] Bounded and monotonic sequences. Cauchy's convergence criterion. Series of non-negative terms, Comparison test, Cauchy's integral test, Cauchy's root test, Ratio test, Raabe's tests, Logarithmic tests. Alternating series. Leibnitz's test. Absolute and conditional convergence. Absolute and conditional convergence of series of real and complex terms, rearrangement of series [Appendix (Chapter 20)] Unit-2 Continuity of functions of single variable, sequential continuity. Properties of continuous functions. Uniform continuity, chain rule of differentiability. Mean value theorems and their geometrical interpretations. Darboux's intermediate value theorem for derivatives. Unit-3 Limit and continuity of functions of two variables, Partial differentiation, Change of variables, Euler’s Theorem on homogeneous functions, Taylor’s theorem for functions of two variables, Jacobians. Unit-4 Envelopes, Evolutes, Maxima and Minima of functions of two variables. Lagrange's multiplier method. Beta and Gamma functions. Unit-5 Double and triple integrals, volumes and surfaces of solids of revolution, Dirichlet's integrals, change of order of integration in double integrals.
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.
ADVANCED CALCULUS & PARTIAL DIFFERENTIAL EQUATIONS SUCCESS KEY-(IN HINDI)
Unit-I– 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