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Multivariable and Vector Calculus: An Introduction

Sarhan M. Musa and David A. Santos
Publisher: 
Mercury Learning and Information
Publication Date: 
2015
Number of Pages: 
420
Format: 
Hardcover
Price: 
79.95
ISBN: 
9781936420285
Category: 
Textbook
[Reviewed by
P. N. Ruane
, on
10/14/2015
]

This book didn’t evolve from the usual process of joint authorship, because the original manuscript was written by the late David Santos (a mathematician) and subsequently converted to into a book by Sarhan Musa (an engineer). The treatment is quite different to most other books on this subject.

The first notable difference is the book’s overall structure. The first third consists of an introduction to vectors, parametric curves and surfaces. The material pertaining to the book’s title is then covered in next two chapters (differentiation and integration). The final third consists of appendices on Maple and MATLAB, answers to odd numbered exercises, a table of useful formulae and extensive bibliography.

One notable strength of the book is the emphasis on computer illustrations, tutorials and exercises using the aforementioned software. It eases the reader into use of this technology by initially employing it for very easy exercises like finding the norm of a vector in the plane. It eventually extends to programming in MATLAB for plotting curves and surfaces in 3-space. Another salient feature is the large number of examples and exercises (the majority of which are routine).

On the negative side, there has been a lack of editorial scrutiny so that there are many typos (or even malapropisms). The preface, for example, refers to the introduction of ‘differentiation forms’, which are later used to state ‘Strokes theorem’ (this misspelling occurs at various points in the text). Later on, and again in several places, there is reference to sets such as \(R =\{(x,y) \in \mathbb{R}^3 \colon x\geq 0,y\geq 0,z\geq 0 \text{ etc.}\}\). And I could go on.

And, as a matter of personal taste, I don’t like the introduction to vectors, which begins with the definition ‘A vector \(\vec{a}\in\mathbb{R}^2\) is a codification of movement of a bi-point’, where a bi-point is a pair \([r,s]\) with \(r = (a, b)\) and \(s = (c, d)\). But, apart from such arcane terminology, good foundations are laid for applications of vector methods to solid and plane geometry and then to curves and surfaces in 3-space. This chapter is also enlivened by less familiar ideas like the ‘surveyors’ theorem’, the Cavalieri principle and the Pappus-Guldin rule.

Coverage of multivariable differentiation includes much of what is expected for an undergraduate course. Bypassing the concept of differential, and relying mainly on the concept of the ‘Jacobi’, there are topics such as the chain rule, Lagrange multipliers and differentiation under an integral sign etc. And yet, given the extensive amount of time spent on vector geometry in the first chapter, the treatment of differentiation is surprisingly non-geometric (no reference to tangent planes or numerical approximation)

Discussion of integration is preceded by the non-intuitive definition of an elementary \(k\)-form as being the \(x_{j_1},x_{j_2},\dots x_{j_k}\) component of the signed \(k\)-parallelotope spanned by a set of \(k\) vectors in \(\mathbb{R}^n\) — which seems unnecessary because integration is then mainly carried out on manifolds of low dimension (one, two and three). Anyway, the payoff is much subsequent material on closed and exact forms, surface integrals, change of variables and, of course, the theorems of Green, Stokes and Gauss.

So, if only for use as a source of problem-solving exercises, this book is worth a look.


Fifty years ago, Peter Ruane was introduced to this area of mathematics via Richard Courant’s (manifold-free) two volume work on Differential and Integral Calculus (he lent them out and never got them back). 

Chapter 1: Vectors and Parametric Curves

Chapter 2: Differentiation

Chapter 3: Integration

Appendix A: Maple

Appendix B: MATLAB

Appendix C: Answers To Odd-Numbered Exercises

Appendix D: Formulas.