Div, Grad, Curl, and All That: An Informal Text on Vector Calculus [ペーパーバック]

This is about as tame a book on vector calculus as you could ever hope to meet, which is part of the reason it's been so popular for so long. It's very easy to read (as far as math texts go), it has many simple but effective illustrations, it has ample exercises (most of which have solutions in the back), and it avoids excessive formalism, instead focusing on the nuts-and-bolts of vector calculus as it most commonly arises in electrostatics, for example.

Math majors will not be so enamored of this book, simply because of its heuristic approach (hence the word "informal" in the subtitle) and its close ties with applications, which it uses as motivation. Moreover, Schey does not develop differential forms or exterior calculus, which logically subsume and extend the material in this book (at the expense of far greater abstraction, which the majority of engineering and science students will prefer to avoid or at least delay). Instructors, if you teach electrostatics or fluid dynamics, you may wish to consider having this as a supplementary text for your students. It's such a clear and helpful little book your students will really appreciate it. (But, you already knew that.)

Bottom line for engineering and science students: You need to know this material, and it simply won't get any easier than this. Don't wait for the audio edition!

My first impression of Schey's book is that it would make a great first course in vector calculus. In fact, I recommend it for that purpose. It will also be very useful for the student enrolled in a class on vector calculus, who wants a secondary reference text to help expand concepts. Schey's approach will appeal to physicists and engineers, with it's intuitive, visual style. Schey uses electric fields as the motivating challenge for developing equations that use the divergence, gradient, and curl, and he uses chapter 1 to develop virtually all the physical concepts needed to follow the derivations. For prerequisites, you should have at least one semester of calculus, and it will help to have a little understanding about electromagnetism, as well (a high school level will be more than adequate for this purpose).

Schey's book also makes a great refresher text (that's why I bought it). If you've had vector calculus in college, you'll be able to read this book in a week or so. It's nicely illustrated, and has problems at the end of each chapter that are strategically designed to extend concepts brought out in the text (solutions to most of the problems appear at the end of the text).

The book's organization is pretty simple, with four sections/chapters. The first is a basic introduction that describes the notion of a vector field and some basic concepts in electrostatics. True to the overall theme throughout the text, Schey uses simple, intuitive explanations and drawings that are especially applicable for beginning students.

The second section introduces surface integrals and divergence. As he does in the remaining chapters, Schey develops equations in Cartesian, spherical, and cylindrical coordinate systems (though he sometimes leaves some of these as exercises for the student). He also summarizes them at the end of the book. In addition to giving the functional, coordinate-dependent form, Schey also shows how the operators are limits that exist as physical entities, independent of any particular coordinate system. For example, Schey summarizes divergence as the limit, as the volume goes to zero, of the flux of the vector field through a surface, divided by the volume enclosed by the surface (see page 37). Beginning texts don't always make this clear, resulting in some students failing to understand divergence (for example) as anything more than the equation that describes it in Cartesian coordinates. But Schey artfully incorporates this more general understanding as part of his clear and intuitive style of teaching.

The third section is about line integrals, the Curl, and Stokes' theorem. The approach is intuitive, with a minimum of formal mathematics, and abundant, clear, diagrams that greatly help to illustrate principles. As with divergence, Schey provides the mathematical form for Curl in three different coordinate systems, as well as the general description (independent of coordinate system): curl is the limit of circulation to area, in the limit, as the area tends to zero.

The fourth, and final section deals with the Gradient. In keeping with the general theme of deriving the mathematical tools to calculate the electric field, Schey summarizes the relationship between the Curl of the vector field, the vector field as the gradient of a scalar function, and the line integral around a closed path of the dot product between the tangent and the vector field. He also extends the notion of the gradient operator to that of the Laplacian, and discusses Poisson's and Laplace's equations. As with the other chapters, Schey makes a point of endowing his explanations with intuitive and visual explanations, explaining that "the gradient of a scalar function F(x,y,z) is a vector that is in the direction in which [the scalar function] F undergoes the greatest rate of increase and that has magnitude equal to the rate of increase in that direction."

I really enjoyed reading this book. Having graduated from university over 20 years ago, I'm not as quick to recall this stuff, so I value a concise book with visual, intuitive, and ready explanations.