Multivariate Calculus and Geometry

Abstract

The importance assigned to accuracy in basic mathematics courses has, initially, a useful disciplinary purpose but can, unintentionally, hinder progress if it fosters the belief that exactness is all that makes mathematics what it is. Multivariate calculus occupies a pivotal position in undergraduate mathematics programmes in providing students with the opportunity to outgrow this narrow viewpoint and to develop a flexible, intuitive and independent vision of mathematics. This possibility arises from the extensive nature of the subject. Multivariate calculus links together in a non-trivial way, perhaps for the first time in a student’s experience, four important subject areas: analysis, linear algebra, geometry and differential calculus. Important features of the subject are reflected in the variety of alternative titles we could have chosen, e.g. ‘‘Advanced Calculus’’, ‘‘Vector Calculus’’, ‘‘Multivariate Calculus’’, ‘‘Vector Geometry’’, ‘‘Curves and Surfaces’’ and ‘‘Introduction to Differential Geometry’’. Each of these titles partially reflects our interest but it is more illuminating to say that here we study differentiable functions, i.e. functions which enjoy a good local approximation by linear functions. The main emphasis of our presentation is on understanding the underlying fundamental principles. These are discussed at length, carefully examined in simple familiar situations and tested in technically demanding examples. This leads to a structured and systematic approach of manageable proportions which gives shape and coherence to the subject and results in a comprehensive and unified exposition. We now discuss the four underlying topics and the background we expect— bearing in mind that the subject can be approached with different levels of mathematical maturity. Results from analysis are required to justify much of this book, yet many students have little or no background in analysis when they approach multivariate calculus. This is not surprising as differential calculus preceded and indeed motivated the development of analysis. We do not list analysis as a prerequisite, but hope that our presentation shows its importance and motivates the reader to study it further.