Miroslav Lovrić’s text Vector Calculus provides a comprehensive introduction to the theory of functions of several variables. As Lovrić describes in the text’s introduction:
Important concepts of calculus of real-valued functions of one variable (limit, continuity, derivative, differentiability, integral) are generalized to functions of several variables and to vector-valued functions. Attempts to generalize the definite integral result in a construction of path and surface integrals. Classical integration theorems of Green, Gauss, and Stokes, which relate the (generalized) concepts of derivative and integral, preserve the spirit of the Fundamental Theorem of Calculus for real-valued functions of one variable.
I imagine that this book will be adopted widely, because it is both well-written and flexible. Depending on the backgrounds, abilities, and interests of the students in the class, the material can be presented at different levels with different goals in mind. The text could be used for a theoretical course geared towards math majors, a more conceptual course geared towards engineering majors, or a general course geared towards a mix of students who have completed the two-semester sequence of calculus of real-valued functions of one-variable.
In reading through the text, I got the distinct impression that Lovrić is a good teacher, one who understands balance in teaching and the importance of using a variety of approaches (algebraic, numeric, and geometric). He meets students where they are, making connections between the material to be presented and the material covered in previous sections or prerequisite courses. For instance, before addressing optimization of functions of two variables, he reminds students of the one variable situation. He sets the stage by quickly reviewing the Extreme Value Theorem, Fermat’s Theorem, and the Second Derivative Test.
Lovrić also makes efforts to create connections between the abstract world of mathematics and the real world around us. Before discussing surface integrals, he presents a section titled “World of Surfaces” in which he explores parameterizations of some famous surfaces: helicoids, catenoids, and hyperbolic paraboloids. After providing the parametric equations for these surfaces, he makes concrete connections between these algebraic objects and actual physical objects. An Archimedes screw is a helicoid, he explains; the soap film surface created by two rings is a catenoid, and the roof of the Catalano House in Raleigh, North Carolina is a hyperbolic paraboloid. He provides pictures of these objects along side the computer-generated parameterizations. Not only do these connections make abstract mathematical objects real for the students, they stress that mathematics is relevant and important.
More generally, I was pleased to see that Lovrić recognizes the value of visualization in the teaching of vector calculus. The book is full of graphs and pictures that make the material more accessible to students. (Just for fun, I randomly opened the book 30 times to find a total of 37 pictures. The book seems to have fewer pictures and diagrams near the end of the text, however.)
How does the book present the theory? As Lovrić explains: “In order to keep the text flowing, proofs of several theorems that are technical and not really revealing, such as differentiation theorems, have been moved to Appendix A at the back of the book.” However, Lovrić also includes proofs throughout the text. Sometimes he presents an “intuitive proof” or an “idea of a proof”, and other times he presents complete proofs. Maintaining balance and a respect for the conceptual, he illuminates the theory with numerous examples and discussions, creating a presentation that is both rigorous and engaging.
My only criticism is that the text has few examples from the social sciences. While the book is rich with applications and examples from physics and engineering, there are not as many examples illustrating the relevance of vector calculus to economics (and I mention this only because I usually have a significant number of economics students in my multivariable calculus classes.) Of course, much of vector calculus was invented to describe physical phenomena, so I can understand the tendency for an author to favor the physical world. Nonetheless, social scientists also stand to gain a great deal from a good knowledge of vector calculus.
Criticism aside, I am a big fan of this textbook. The material is presented clearly and supplemented with helpful visualizations, insightful examples, and useful applications. I believe that students will find this text to be engaging and easy to read.
Judy Holdener (firstname.lastname@example.org) is an associate professor of mathematics and the John B. McCoy Distinguished Teaching Chair at Kenyon College in Ohio. Her mathematical interests include number theory, algebra, and dynamical systems, and her personal interests include painting and the arts. As a firm believer of the importance of visualization in mathematics and the teaching of mathematics, vector calculus is one of her favorite courses to teach.