Mathematical Analysis: Functions of Several Real Variables and Applications

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 effortlessly determine 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.

這本書是由該領域的專家講師撰寫的數學分析教科書。除了傳統的多變數函數微分和積分工具、度量空間、常微分方程、隱函數等，這本教科書還提供了深入研究某些主題的機會：其中包括Ascoli-Arzelà定理、R n中凸函數的正則性、L p空間和絕對連續函數等，這些主題在現代數學分析中非常重要。其他例子包括Weierstrass定理關於連續函數的多項式逼近或Peano存在定理（通常只有存在性，沒有唯一性）對於非線性常微分方程和系統在一般假設下的應用。

內容以初級方式進行討論，並在後續階段從多個更深入的角度檢視某些主題。主題的靈活組織有助於教師輕鬆確定在講座中應該呈現的部分以及何時停止。作者認為，任何教科書只能在一定程度上對講座課程的成功做出貢獻，而講師的選擇在這方面起著決定性的作用。

本書針對在理學、工程學、經濟學和建築學等科學、工程、經濟和建築學院修讀數學分析課程的研究生或本科榮譽學生。

​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.

Nicola Fusco是那不勒斯大學"費代里科二世"的數學分析全職教授，也是意大利林切學院的成員。他獲得了1994年意大利數學聯盟（UMI）的Caccioppoli獎。他的研究主要集中在變分計算、偏微分方程的正則性理論、對稱化問題和等周不等式。他曾擔任澳大利亞國立大學坎培拉分校、卡內基梅隆大學匹茲堡分校、赫里奧特-瓦特大學愛丁堡分校、牛津大學、慕尼黑工業大學和約伯斯基大學的客座教授。

Paolo Marcellini是佛羅倫薩大學的名譽數學分析教授。他的研究興趣包括變分計算和偏微分方程的正則性理論。他曾擔任佛羅倫薩大學理學院院長和GNAMPA（國家數學分析、概率及其應用組）的主席。他曾擔任加州大學伯克利分校、法國學院、普林斯頓高等研究院、卡內基梅隆大學非線性分析中心、牛津大學數學研究所、德克薩斯大學奧斯汀分校和斯德哥爾摩Mittag-Leffler研究所的客座教授。

Carlo Sbordone是那不勒斯大學"費代里科二世"的名譽數學分析教授，也是意大利林切學院的成員，曾擔任意大利數學聯盟（UMI）的主席。他的研究興趣包括變分計算、Sobolev映射和函數空間。他曾擔任比薩諾馬爾超常規學校、法國學院、蘇黎世大學數學研究所、卡內基梅隆大學非線性分析中心、加州大學伯克利分校、牛津大學數學研究所和赫爾辛基大學的客座教授。