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 possi- bility 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.