Derivatives and Integrals of Multivariable Functions

This text is appropriate for a one-semester course in what is usually called ad vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.

List of contents

1 Differentiability of Multivariate Functions.- 1.1 Differentiability.- 1.2 Derivatives and Partial Derivatives.- 1.3 The Chain Rule.- 1.4 Higher Derivatives.- 2 Derivatives of Scalar Functions.- 2.1 Directional Derivatives and the Gradient.- 2.2 The Mean Value Theorem.- 2.3 Extreme Values and the Derivative.- 2.4 Extreme Values and the Second Derivative.- 2.5 Implicit Scalar Functions.- 2.6 Curves, Surfaces, Tangents, and Normals.- 3 Derivatives of Vector Functions.- 3.1 Contractions.- 3.2 The Inverse Function Theorem.- 3.3 The Implicit Function Theorem.- 3.4 Lagrange's Method.- 4 Integrability of Multivariate Functions.- 4.1 Partitions.- 4.2 Integrability in a Box.- 4.3 Domains of Integrability.- 4.4 Integrability and Sets of Zero Volume.- 5 Integrals of Scalar Functions.- 5.1 Fubini's Theorem.- 5.2 Properties of Integrals.- 5.3 Change of Variable.- 5.4 Generalized Integrals.- 5.5 Line Integrals.- 5.6 Surface Integrals.- 6 Vector Integrals and the Vector-Field Theorems.- 6.1 Integrals of the Tangential and Normal Components.- 6.2 Path-Independence.- 6.3 On the Edge: The Theorems of Green and Stokes.- 6.4 Gauss's Theorem.- Solutions to Exercises.- References.

Summary

This work provides a systematic examination of derivatives and integrals of multivariable functions. The approach taken here is similar to that of the author’s previous "Continuous Functions of Vector Variables": specifically, elementary results from single-variable calculus are extended to functions in several-variable Euclidean space. Topics encompass differentiability, partial derivatives, directional derivatives and the gradient; curves, surfaces, and vector fields; the inverse and implicit function theorems; integrability and properties of integrals; and the theorems of Fubini, Stokes, and Gauss. Prerequisites include background in linear algebra, one-variable calculus, and some acquaintance with continuous functions and the topology of the real line.
 

Written in a definition-theorem-proof format, the book is replete with historical comments, questions, and discussions about strategy, difficulties, and alternate paths. "Derivatives and Integrals of Multivariable Functions" is a rigorous introduction to multivariable calculus that will help students build a foundation for further explorations in analysis and differential geometry.

Additional text

"Guzman . . . offers a well-crafted treatment of this standard material, intentionally more advanced than in the most widely used books. . . . With that emphasis on the theoretical aspects of the subject, the book will appeal . . . to students intending ultimately to pursue graduate work in mathematics . . . Recommended."
—CHOICE
"The book under review presents a systematic introduction to the calculus of real functions of several variables…. The presentation of the material is very clear throughout. Proofs are carefully worked out, theorems are illustrated by examples, and exercises at the end of each section will help to master the calculus. The book is highly recommendable to students to build a foundation for further studies in analysis and differential geometry."
—ZENTRALBLATT MATH
"The book is evidently intended for undergraduate courses and opens the way for abstract generalization and exploring analysis, differential geometry and maybe also physics. The book is written very carefully and rigorously. The style is easily understandable with many comments supporting understanding. Problems with solutions at the end of the book are included.  This book together with the above mentioned volume of A. Guzman is a nice source for a one year course at the undergraduate level." ---Mathematica Bohemica
 

Report

"Guzman . . . offers a well-crafted treatment of this standard material, intentionally more advanced than in the most widely used books. . . . With that emphasis on the theoretical aspects of the subject, the book will appeal . . . to students intending ultimately to pursue graduate work in mathematics . . . Recommended."
-CHOICE
"The book under review presents a systematic introduction to the calculus of real functions of several variables.... The presentation of the material is very clear throughout. Proofs are carefully worked out, theorems are illustrated by examples, and exercises at the end of each section will help to master the calculus. The book is highly recommendable to students to build a foundation for further studies in analysis and differential geometry."
-ZENTRALBLATT MATH
"The book is evidently intended for undergraduate courses and opens the way for abstract generalization and exploring analysis, differential geometry and maybe also physics. The book is written very carefully and rigorously. The style is easily understandable with many comments supporting understanding. Problems with solutions at the end of the book are included.  This book together with the above mentioned volume of A. Guzman is a nice source for a one year course at the undergraduate level." ---Mathematica Bohemica
 

Product details

• Authors: A. Guzman, Alberto Guzman
• Publisher: Springer, Basel
• Languages: English
• Product format: Paperback / Softback
• Released: 09.10.2003
• EAN: 9780817642747
• ISBN: 978-0-8176-4274-7
• No. of pages: 319
• Weight: 480 g
• Illustrations: X, 319 p. 30 illus.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Analysis, B, measure theory, Mathematics and Statistics, Real Functions, Numerical analysis, Manifolds (Mathematics), Functions of real variables, Calculus & mathematical analysis, Integral calculus & equations, Analysis (Mathematics), Mathematical analysis, Measure and Integration, Global analysis (Mathematics), Global Analysis and Analysis on Manifolds