Functions of Several Real Variables

Functions of Several Real Variables

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.

Author: Martin A. Moskowitz

Publisher: World Scientific

ISBN: 9789814299268

Category: Mathematics

Page: 716

View: 167

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. 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.
Categories: Mathematics

Several Real Variables

Several Real Variables

II (Interscience, London, 1937) A. Devinatz, Advanced Calculus (Holt, Rinehart and Winston, New York, 1968) C.H. Edwards, Advanced Calculus of Several Variables (Dover, New York, 1994) H.H. Edwards, Advanced Calculus, a Differential ...

Author: Shmuel Kantorovitz

Publisher: Springer

ISBN: 9783319279565

Category: Mathematics

Page: 307

View: 716

This undergraduate textbook is based on lectures given by the author on the differential and integral calculus of functions of several real variables. The book has a modern approach and includes topics such as: •The p-norms on vector space and their equivalence •The Weierstrass and Stone-Weierstrass approximation theorems •The differential as a linear functional; Jacobians, Hessians, and Taylor's theorem in several variables •The Implicit Function Theorem for a system of equations, proved via Banach’s Fixed Point Theorem •Applications to Ordinary Differential Equations •Line integrals and an introduction to surface integrals This book features numerous examples, detailed proofs, as well as exercises at the end of sections. Many of the exercises have detailed solutions, making the book suitable for self-study. Several Real Variables will be useful for undergraduate students in mathematics who have completed first courses in linear algebra and analysis of one real variable.
Categories: Mathematics

Functions Of Several Real Variables

Functions Of Several Real Variables

97 3 Differential Calculus in Several Variables 103 3.1 Differentiable functions . . . . . . . . . . . . . . . . . . . 103 3.2 Partial and directional derivatives, tangent space . . . . 109 3.3 Homogeneous functions and Euler's equation ...

Author: Martin Moskowitz

Publisher: World Scientific Publishing Company

ISBN: 9789813100916

Category: Mathematics

Page: 732

View: 305

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 such as the Inverse and Implicit Function theorems, Lagrange's multiplier rule, Fubini's theorem, the change of variables formula, Green's, Stokes' and Gauss' theorems are proved in detail and many of them with novel proofs. The authors develop the theory in a logical sequence building one result upon the other, enriching the development with numerous explanatory remarks and historical footnotes. A number of well chosen illustrative examples and counter-examples clarify matters and teach the reader how to apply these results and solve problems in mathematics, the 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, Sard's theorem , the Weierstrass approximation theorem, the Fourier transform, Vector fields on spheres, Brouwer's fixed point theorem, Whitney's embedding theorem, Picard's theorem, and Hermite polynomials are discussed in stared sections.
Categories: Mathematics

On the Differentiability of Functions of Several Real Variables

On the Differentiability of Functions of Several Real Variables

... well known theorem of Lebesgue which states that any monotone function of a single real variable has a finite derivative almost everywhere . The following lemma generalizes this result to functions of several variables . I LEMMA 1 .

Author: James Serrin

Publisher:

ISBN: UOM:39015095244490

Category: Functions of several real variables

Page: 24

View: 619

Categories: Functions of several real variables

Functions of several variables

Functions of several variables

This book is aimed at mathematics students, typically in the second year of a university course.

Author: B. Craven

Publisher: Springer Science & Business Media

ISBN: 9789401093477

Category: Science

Page: 138

View: 826

This book is aimed at mathematics students, typically in the second year of a university course. The first chapter, however, is suitable for first-year students. Differentiable functions are treated initially from the standpoint of approximating a curved surface locally by a fiat surface. This enables both geometric intuition, and some elementary matrix algebra, to be put to effective use. In Chapter 2, the required theorems - chain rule, inverse and implicit function theorems, etc- are stated, and proved (for n variables), concisely and rigorously. Chapter 3 deals with maxima and minima, including problems with equality and inequality constraints. The chapter includes criteria for discriminating between maxima, minima and saddlepoints for constrained problems; this material is relevant for applications, but most textbooks omit it. In Chapter 4, integration over areas, volumes, curves and surfaces is developed, and both the change-of-variable formula, and the Gauss-Green-Stokes set of theorems are obtained. The integrals are defined with approximative sums (ex pressed concisely by using step-functions); this preserves some geometrical (and physical) concept of what is happening. Consequent on this, the main ideas of the 'differential form' approach are presented, in a simple form which avoids much of the usual length and complexity. Many examples and exercises are included.
Categories: Science

Advanced Calculus of Several Variables

Advanced Calculus of Several Variables

Advanced Calculus of Several Variables provides a conceptual treatment of multivariable calculus. This book emphasizes the interplay of geometry, analysis through linear algebra, and approximation of nonlinear mappings by linear ones.

Author: C. H. Edwards

Publisher: Academic Press

ISBN: 9781483268057

Category: Mathematics

Page: 470

View: 310

Advanced Calculus of Several Variables provides a conceptual treatment of multivariable calculus. This book emphasizes the interplay of geometry, analysis through linear algebra, and approximation of nonlinear mappings by linear ones. The classical applications and computational methods that are responsible for much of the interest and importance of calculus are also considered. This text is organized into six chapters. Chapter I deals with linear algebra and geometry of Euclidean n-space Rn. The multivariable differential calculus is treated in Chapters II and III, while multivariable integral calculus is covered in Chapters IV and V. The last chapter is devoted to venerable problems of the calculus of variations. This publication is intended for students who have completed a standard introductory calculus sequence.
Categories: Mathematics

Real Analysis

Real Analysis

Intended as a sequel to a course in single variable analysis, this book builds upon and expands these ideas into higher dimensions. The modular organization makes this text adaptable for either a semester or year-long introductory course.

Author: Miklós Laczkovich

Publisher: Springer

ISBN: 9781493973699

Category: Mathematics

Page: 392

View: 501

This book develops the theory of multivariable analysis, building on the single variable foundations established in the companion volume, Real Analysis: Foundations and Functions of One Variable. Together, these volumes form the first English edition of the popular Hungarian original, Valós Analízis I & II, based on courses taught by the authors at Eötvös Loránd University, Hungary, for more than 30 years. Numerous exercises are included throughout, offering ample opportunities to master topics by progressing from routine to difficult problems. Hints or solutions to many of the more challenging exercises make this book ideal for independent study, or further reading. Intended as a sequel to a course in single variable analysis, this book builds upon and expands these ideas into higher dimensions. The modular organization makes this text adaptable for either a semester or year-long introductory course. Topics include: differentiation and integration of functions of several variables; infinite numerical series; sequences and series of functions; and applications to other areas of mathematics. Many historical notes are given and there is an emphasis on conceptual understanding and context, be it within mathematics itself or more broadly in applications, such as physics. By developing the student’s intuition throughout, many definitions and results become motivated by insights from their context.
Categories: Mathematics

First Steps in Several Complex Variables

First Steps in Several Complex Variables

This equation in the case of several real variables corresponds to the Laplace equation Au = 0. Consequently , the harmonic functions of n real variables correspond in some sense to the affine functions of one real variable .

Author: Marek Jarnicki

Publisher: European Mathematical Society

ISBN: 3037190493

Category: Functions of several complex variables

Page: 359

View: 865

This book provides a comprehensive introduction to the field of several complex variables in the setting of a very special but basic class of domains, the so-called Reinhardt domains. In this way the reader may learn much about this area without encountering too many technical difficulties. Chapter 1 describes the fundamental notions and the phenomenon of simultaneous holomorphic extension. Chapter 2 presents a fairly complete discussion of biholomorphisms of bounded (complete) Reinhardt domains in the two dimensional case. The third chapter gives a classification of Reinhardt domains of existence for the most important classes of holomorphic functions. The last chapter deals with invariant functions and gives explicit calculations of many of them on certain Reinhardt domains. Numerous exercises are included to help the readers with their understanding of the material. Further results and open problems are added which may be useful as seminar topics. The primary aim of this book is to introduce students or non-experts to some of the main research areas in several complex variables. The book provides a friendly invitation to this field as the only prerequisite is a basic knowledge of analysis.
Categories: Functions of several complex variables

Tasty Bits of Several Complex Variables

Tasty Bits of Several Complex Variables

A harmonic function of one complex variable is in some sense a generalization of an affine linear function of one real variable. Similarly, as far as several complex variables are concerned, a pluriharmonic function is the right ...

Author: Jiri Lebl

Publisher: Lulu.com

ISBN: 9780359642250

Category: Mathematics

Page: 184

View: 118

"This book is a polished version of the author’s notes for a course entitled Several Complex Variables. It should be suitable for a semester-long topics course or for self-study as an introduction to the subject. The prerequisites are decent knowledge of vector calculus, basic real analysis, and a working knowledge of complex analysis in one variable. It should be accessible to beginning graduate students after a complex analysis course, and perhaps even very advanced undergraduates. This is enough material for a semester-long course, including quite a few exercises sprinkled throughout the text, all of which the reader should at least be attempting. It is not meant as an exhaustive reference, but simply as a whirlwind tour of several complex variables"--BCcampus website.
Categories: Mathematics

Four papers on functions of real variables

Four papers on functions of real variables

THE PROPERTIES OF SOME CLASSES OF DIFFERENTIABLE FUNCTIONS OF SEVERAL VARIABLES DEFINED IN AN n - DIMENSIONAL REGION V. P. IL'IN INTRODUCTION In the present article , properties falling under the general heading of embedding theorems ...

Author: Kirill Kapitonovich Golovkin Valeri_ Pavlovich Il_in Vsevolod Alekseevich Solonnikov

Publisher: American Mathematical Soc.

ISBN: 0821896547

Category:

Page:

View: 640

Categories: