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© 1981

Functions of several variables

Book

Table of contents

  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

About this book

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

  1. 1.University of MelbourneAustralia

Bibliographic information