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Basic Insights in Vector Calculus

Terrance J. Quinn, Zine Broudhraa, and Sanjay Rai
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
John Ross
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This book was created with a clear purpose to serve as a complementary text to a vector calculus course. It offers insight and motivation into the major theorems of undergraduate single-variable and multivariable calculus. In this, the text is successful – working from physical principles and complete with many pictures, the text demystifies the “big theorems” that arise in the calculus sequence. As a result, the text is a worthwhile and engaging read for both undergraduate and graduate students looking to better understand the results that appear at the end of a traditional Calculus 3 course. It would also be recommended to a professor of Calculus 3 who is looking for inquiry-based approaches or thought experiments to use in their course.
The main premise of the book is that traditional texts tend to do many things well when introducing the big theorems of Vector Calculus – they provide applications, examples for developing symbolic and computational techniques, and offer (at least partial) proofs – but they tend to do a poor job of properly providing motivational insight. Insight, the authors argue, is crucial. Without insight, being asked to read a proof is being asked to read a solution to a problem that has not yet been formed. And without insight, symbolic and computational mastery is difficult to achieve. What’s more, students in this course come from a number of different possible majors (including mathematics, physics, and engineering), and so insight should be accessible to students of multiple backgrounds.
Once the weakness of traditional textbooks has been identified, the authors seek to rectify this by developing “questions and key insights needed to reach basic understanding of the fundamental theorems of vector calculus.” They achieve this by drawing from the historical development of mathematical physics that led to the results initially. The result is a carefully crafted series of inquiry-based narratives and thought experiments that offer insight, build up intuition, and lead the reader to the major theorems of Vector Calculus.
The text is broken up into sections. Section 1 reviews key foundational material from Calculus, building to an understanding of the FTC, the (single variable and multivariable) chain rule, and the cross product, among other essentials. This helps to lay some groundwork, both in basic Calculus concepts and in how the rest of the text will be structured. Section 2 focuses on fluid motion in 2 dimensions, with subsections dedicated to fluid circulation, Green’s Theorem, and 2-d flux, and divergence. Section 3 then focuses on fluid motion in 3 dimensions, extrapolating from the previous work in 2-d to explore flux, divergence, Stokes’ Theorem, and relative change in volumes and in increments. Well-crafted pictures are used extensively throughout the text, and each subsection ends by posing exploratory problems based on the preceding narrative. 
The text concludes with a supplemental section meant to advocate for the importance of insight in the mathematical process. This section is not focused on calculus content, but rather on the process in which we, as mathematicians and students, learn and create mathematical meaning. I suspect that this section is not strictly necessary, as those who seek to use this text will already understand the importance of developing mathematical insight. Nevertheless, this concluding section certainly does not hurt, and does serve to reinforce the authors’ argument in favor of teaching in a pedagogically sound manner based on our understanding of the process by which mathematicians learn mathematics. 
In summary, this text delivers on its promise to provide motivation and insight to the major theorems of Vector Calculus. While it would not serve as a standalone textbook, Basic Insights in Vector Calculus has something to offer for those looking to develop a deeper understanding of the major results that serve as a capstone of the undergraduate calculus sequence. 


John Ross is an Assistant Professor of Mathematics at Southwestern University