Vector Calculus

Reviewed by Yaping Liu, Professor, Pittsburg State University on 1/12/23

Comprehensiveness rating: 4 see less

This book contains about enough material for a one semester multivariable calculus or a beginning vector calculus course. It is relatively easy to read and follow. At the end of each section a fair number of exercises are provided, which are divided into 3 categories, A, B, C, roughly based on the level of difficulty. A number of routine examples are provided to demonstrate mathematical concepts and basic techniques in calculation. Color-coded boxes are used in the text to highlight the definitions, theorems, and other important results. Answers and hints to selected exercises are provided in Appendix A toward the end of the book. A useful index is also included.

Content Accuracy rating: 5

This is a neatly organized little book on vector calculus. It is well written with mathematical accuracy. The proofs for some theorems are provided, while some others are left as exercises.

Relevance/Longevity rating: 4

This book is concise. The content is carefully filtered. Many relevant topics are omitted, only briefly treated, or left as exercises. The book is for those who share a similar preference over the topics as the author. An instructor will have limited choices. Or one can use the book by selecting the topics one likes and supplements it with content found elsewhere. I personally prefer that it contains some more advanced topics, such as the implicit function theorem and the Taylor series expansion of multivariable functions, and more involved real world examples in physical sciences so that it can also be used as a vector calculus textbook following the calculus sequence. There is such a need among senior or beginning graduate level STEM students. Expansion, if desired, can be done in future updates.

Clarity rating: 5

This book is nicely structured. Mathematical concepts are sufficiently explained. Fully worked out examples are given as appropriate. Good quality figures are generated and included to illustrate the ideas. Very few long examples with tedious calculations are included. A good student should be able to understand most of the content through self-study.

Consistency rating: 5

The book is carefully written. The notations and terms used are consistent throughout the book. The examples, definitions, theorems, and figures are numbered separately and sequentially in each chapter. Math styles and different fonts are used appropriately and consistently.

Modularity rating: 4

The main content is divided into four chapters. Each chapter is then divided into a number of sections. An instructor can easily organize the content into units to suit the flow in their class. Reorganization would be relatively hard due partly to the logical dependence of the topics, which is typical for math textbooks, and partly to the fact that this book is already lean.

Organization/Structure/Flow rating: 5

The organization is clear. It follows a logical sequence of the topics from vectors in Euclidean space to vector-valued functions, then functions of several variables, and finally line and surface integrals. The content is arranged basically the same way as in other standard books on this subject.

Interface rating: 4

A bibliography is compiled. Further explanation of certain ideas can be found in a number of books referenced in the foot notes. Otherwise the book is self-contained. There are no hyperlinks. Images and charts are properly formatted with little distortion. The flow of the content is smooth and clear.

Grammatical Errors rating: 5

It is well written with clear and straightforward English. I didn’t find any conspicuous grammatical errors.

Cultural Relevance rating: 5

This book contains no offensive material. There is little evidence the author tried to come up with real world examples that are inclusive of races, ethnicities, and background. But by being culturally neutral, as most mathematics books are, the book is inclusive by default.

Comments

The book's main strength is also its main weakness. By being focused on the selected topics it covers with little extra details or digression on other topics, it limits the way the book can be used. It can serve it's intended purpose very well, i.e., a textbook for calculus 3 or introductory vector calculus. To go beyond this level, it is necessary to supplement it with more advanced materials.