Mathematical Analysis

This work is a textbook on Mathematical Analysis written by expert lecturers in the field. This textbook, other than the classical differentiation and integration tools for functions of several real variables, metric spaces, ordinary differential equations, implicit function and so on, also provides opportunities to go deeper into certain topics: among them, the Ascoli-Arzelà theorem, the regularity of convex functions in R^n, L^p spaces and absolutely continuous functions, all topics that are paramount in modern Mathematical Analysis. Other instances include the Weierstrass theorem on polynomial approximation of continuous functions or Peano's existence theorem (typically only existence, without uniqueness) for nonlinear ODEs and systems under general assumptions. The content is discussed in an elementary way and, at a successive stage, some topics are examined from several, more penetrating, angles. The agile organization of the subject matter helps instructors to effortlesslydetermine which parts to present during lectures and where to stop. The authors believe that any textbook can contribute to the success of a lecture course only to a point, and the choices made by lecturers are decisive in this respect.

The book is addressed to graduate or undergraduate honors students in Mathematics, Physics, Astronomy, Computer Science, Statistics and Probability, attending Mathematical Analysis courses at the Faculties of Science, Engineering, Economics and Architecture.

Auteur

Nicola Fusco is Full Professor of Mathematical Analysis at University of Naples "Federico II" and member of Accademia dei Lincei. He was awarded the 1994 Caccioppoli Prize of the Italian Mathematical Union (UMI). His research revolves around calculus of variations, regularity theory for partial differential equations, symmetrization problems and isoperimetric inequalities. He was visiting professor at Australian National University, Canberra; Carnegie Mellon University, Pittsburgh; Heriot-Watt University, Edinburgh; University of Oxford; Technische Universität, München and University of Jyväskylä. Paolo Marcellini is Emeritus Professor of Mathematical Analysis at University of Florence. His research interests are in calculus of variations and regularity theory for partial differential equations. He was Dean of the Faculty of Sciences at University of Florence and President of GNAMPA (National Group for Mathematical Analysis, Probability and their Applications). He was visiting professor at University of California, Berkeley; Collège de France, Paris; Institute for Advanced Study, Princeton; Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh; Mathematical Institute, University of Oxford; University of Texas, Austin and Institut Mittag-Leffler, Stockholm. Carlo Sbordone is Emeritus Professor of Mathematical Analysis at University of Naples "Federico II", member of Accademia dei Lincei and was President of the Italian Mathematical Union (UMI). His research interests regard calculus of variations, Sobolev maps and function spaces. He was visiting professor at Scuola Normale Superiore in Pisa; Collège de France, Paris; Institut für Mathematik, Universität Zürich; Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh; University of California, Berkeley; Mathematical Institute, University of Oxford and University of Helsinki.

Contenu

1 Sequences and Series of Functions.- 2 Metric Spaces and Banach Spaces.- 3 Functions of Several Variables.- 4 Ordinary Differential Equations.- 5 Linear Differential Equations.- 6 Curves and Integrals Along Curves.- 7 Differential One-Forms.- 8 Multiple Integrals.- 9 The Lebesgue Integral.- 10 Surfaces and Surface Integrals.- 11 Implicit Functions.- 12 Manifold in Rn and k-Forms.