Functions of Several Real Variables

Description

This book begins with the basics of the geometry and topology of Euclidean space and continues with the main topics in the theory of functions of several real variables including limits, continuity, differentiation and integration. All topics and in particular, differentiation and integration, are treated in depth and with mathematical rigor. The classical theorems of differentiation and integration are proved in detail and many of them with novel proofs. The authors develop the theory in a logical sequence building one theorem upon the other, enriching the development with numerous explanatory remarks and historical footnotes. In this title, a number of well chosen illustrative examples and counter-examples clarify the theory and teach the reader how to apply it to solve problems in mathematics and other sciences and economics. Each of the chapters concludes with groups of exercises and problems, many of them with detailed solutions while others with hints or final answers. More advanced topics, such as Morse's lemma, Brouwer's fixed point theorem, Picard's theorem and the Weierstrass approximation theorem are discussed in stared sections. Preface and Acknowledgments Notations Contents 1 Basic Features of Euclidean Space, Rn 2 Functions on Euclidean Spaces 3 Differential Calculus in Several Variables 4 Integral Calculus in Several Variables 5 Change of Variables Formula, Improper Multiple Integrals 6 Line and Surface Integrals 7 Elements of Ordinary and Partial Differential Equations 8 An Introduction to the Calculus of Variations Appendix A Countability and Decimal Expansions Appendix B Calculus in One Variable Appendix C Uniform Convergence Appendix D Linear Algebra Bibliography Index Página em branco