Continuous Functions of Vector Variables

Alberto Guzman

ISBN 10: 0817642730 / ISBN 13: 9780817642730

Verlag: Birkhäuser Boston, Birkhäuser Boston Jul 2002, 2002

Beschreibung

Neuware -This text is appropriate for a one-semester course in what is usually called ad vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding continuous functions of one real variable. We begin by considering the function f(x, y, . ) of multiple variables as a function of the single vector variable (x, y, . ). It turns out that most of the n treatment does not need to be limited to the finite-dimensional spaces R , so we will often place ourselves in an arbitrary vector space equipped with the right tools of measurement. We then proceed much as one does with functions on R. First we give an algebraic and metric structure to the set of vectors. We then define limits, leading to the concept of continuity and to properties of continuous functions. Finally, we enlarge upon some topological concepts that surface along the way. A thorough understanding of single-variable calculus is a fundamental require ment. The student should be familiar with the axioms of the real number system and be able to use them to develop elementary calculus, that is, to define continuous junction, derivative, and integral, and to prove their most important elementary properties. Familiarity with these properties is a must. To help the reader, we provide references for the needed theorems.Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, 14197 Berlin 220 pp. Englisch.

Bestandsnummer des Verkäufers 9780817642730

Bibliografische Details

• Titel: Continuous Functions of Vector Variables
• Verlag: Birkhäuser Boston, Birkhäuser Boston Jul 2002
• Erscheinungsjahr: 2002
• Sprache: Englisch
• ISBN-10: 0817642730
• ISBN-13: 9780817642730
• Einband: Taschenbuch
• Zustand: Neu
• Abmessungen: 235B x 155 H x 13 T Millimeter Breite: 235 mm, Höhe: 155 mm, Tiefe: 13 mm
• Gewicht: 341 Gramm

Über diesen Titel

Inhaltsangabe

This is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. The author covers boundedness, extreme values, and uniform continuity of functions, along with connections between continuity and topological concepts such as connectedness and compactness. The order of topics mimics the order of development in elementary calculus, with analogies and generalizations from such familiar ideas as the Pythagorean theorem.

Críticas

"The book is devoted to mathematical analysis in multi-dimensional spaces, including infinite-dimensional ones...One of the important characteristics of the text is the topological approach. ...It is worth mentioning that all these ideas are presented in conformity with the situation, i.e. they are strictly related to normed spaces and no unnecessary abstract setting is used. ... The language is very clear and sometimes very ‘fresh’ if compared with classical textbooks, so that it may better appeal to the reader. There are many exercises, and the book concludes with a chapter containing solutions."

―MATHEMATICAL REVIEWS

"This is a textbook on calculus of several variables. It covers algebraic and metric structure of the Euclidean space, convergence, basic properties of continuous functions and topology in normed spaces.... In my opinion, this topological approach is one of the advantages of the book under review. It is written very clearly and contains numerous examples and instructive pictures. Each section is endowed with a set of exercises, and the book is concluded with solutions to these exercises. This is a nice and user-friendly textbook which can be recommended for a one-semester course in multivariable calculus."

―ZENTRALBLATT MATH

"The presentation is detailed and clear. The leisurely discursive style adopted by the author will be appreciated by beginners. The fact that the solution of each and every exercise can be found at the end of the book, considerable enhances its value and makes it suitable also for individual study. This book successfully bridges the gap between elemntary calculus and such higher disciplines as real functions, general topology, and, in particular, functional analysis."

―PUBLICATIONES MATHEMATICAE, DEBRECEN