# Functions of several variables

• B. D. Craven Book

1. Front Matter
Pages i-viii
2. B. D. Craven
Pages 1-12
3. B. D. Craven
Pages 13-40
4. B. D. Craven
Pages 41-66
5. B. D. Craven
Pages 67-134
6. Back Matter
Pages 112-137

### Introduction

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.

### Keywords

Implicit function calculus derivative extrema integral integration maximum minimum

#### Authors and affiliations

• B. D. Craven
• 1
1. 1.University of MelbourneAustralia

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-010-9347-7