# Derivatives and Integrals of Multivariable Functions

• Alberto Guzman
Textbook

1. Front Matter
Pages i-xi
2. Alberto Guzman
Pages 1-32
3. Alberto Guzman
Pages 33-71
4. Alberto Guzman
Pages 73-103
5. Alberto Guzman
Pages 105-134
6. Alberto Guzman
Pages 135-200
7. Alberto Guzman
Pages 201-246
8. Back Matter
Pages 247-319

### Introduction

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.

### Keywords

Derivative Implicit function Measure Theory Multivariable Calculus calculus differential geometry ksa mean value theorem real analysis

#### Authors and affiliations

• Alberto Guzman
• 1
1. 1.Department of MathematicsThe City College of New York, CUNYNew YorkUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-0035-2
• Copyright Information Birkhäuser Boston 2003
• Publisher Name Birkhäuser, Boston, MA
• eBook Packages
• Print ISBN 978-0-8176-4274-7
• Online ISBN 978-1-4612-0035-2
• Buy this book on publisher's site