Mathematical Analysis II

This secondEnglish edition of a very popular two-volume work presents a thorough firstcourse in analysis, leading from real numbers to such advanced topics asdifferential forms on manifolds asymptotic methods Fourier, Laplace, andLegendre transforms elliptic functions and distributions. Especially notablein this course are the clearly expressed orientation toward the naturalsciences and the informal exploration of the essence and the roots of the basicconcepts and theorems of calculus. Clarity of exposition is matched by a wealthof instructive exercises, problems, and fresh applications to areas seldomtouched on in textbooks on real analysis.The maindifference between the second and first English editions is the addition of aseries of appendices to each volume. There are six of them in the first volumeand five in the second. The subjects of these appendices are diverse. They aremeant to be useful to both students (in mathematics and physics) and teachers,who may be motivated by different goals. Some of the appendices are surveys,both prospective and retrospective. The final survey establishes importantconceptual connections between analysis and other parts of mathematics.This second volumepresents classical analysis in its current form as part of a unifiedmathematics. It shows how analysis interacts with other modern fields ofmathematics such as algebra, differential geometry, differential equations,complex analysis, and functional analysis. This book provides a firm foundationfor advanced work in any of these directions.“The textbook of Zorich seems to me the mostsuccessful of the available comprehensive textbooks of analysis formathematicians and physicists. It differs from the traditional exposition intwo major ways: on the one hand in its closer relation to natural-scienceapplications (primarily to physics and mechanics) and on the other hand in a greater-than-usualuse of the ideas and methods of modern mathematics, that is, algebra, geometry,and topology. The course is unusually rich in ideas and shows clearly the powerof the ideas and methods of modern mathematics in the study of particularproblems. Especially unusual is the second volume, which includes vectoranalysis, the theory of differential forms on manifolds, an introduction to thetheory of generalized functions and potential theory, Fourier series and the Fouriertransform, and the elements of the theory of asymptotic expansions. At presentsuch a way of structuring the course must be considered innovative. It wasnormal in the time of Goursat, but the tendency toward specialized courses, noticeableover the past half century, has emasculated the course of analysis, almost reducingit to mere logical justifications. The need to return to more substantive coursesof analysis now seems obvious, especially in connection with the applied characterof the future activity of the majority of students....In my opinion, this course is the best of theexisting modern courses of analysis.”From areview by V.I.ArnoldVLADIMIR A. ZORICH is professor of mathematics at Moscow State University. His areas of specialization are analysis, conformal geometry, quasiconformal mappings, and mathematical aspects of thermodynamics. He solved the problem of global homeomorphism for space quasiconformal mappings. He holds a patent in the technology of mechanical engineering, and he is also known by his book Mathematical Analysis of Problems in the Natural Sciences.

9 Continuous Mappings (General Theory).- 10 Differential Calculus from a GeneralViewpoint.- 11 Multiple Integrals.- 12 Surfaces and Differential Forms in Rn.- 13 Line and SurfaceIntegrals.- 14 Elements of VectorAnalysis and Field Theory.- 15 Integration of Differential Forms onManifolds.- 16 Uniform Convergence andBasic Operations of Analysis.- 17Integrals Depending on a Parameter.- 18Fourier Series and the Fourier Transform.- 19 Asymptotic Expansions.- Topics and Questions for MidtermExaminations.- Examination Topics.-Examination Problems (Series and Integrals Depending on a Parameter).- Intermediate Problems (Integral Calculus ofSeveral Variables).- Appendices: A Series as a Tool (Introductory Lecture).- BChange of Variables in Multiple Integrals.- C Multidimensional Geometry and Functions of a Very Large Number ofVariables.- D Operators of Field Theoryin Curvilinear Coordinates.- E ModernFormula of Newton–Leibniz.- References.-Index of Basic Notation.- Subject Index.- Name Index.