DIV, Grad, Curl, and All That: An Informal Text on Vector Calculus

Simply put, this book provides the best explanation of the gradient, the divergence, and the curl in any book I've seen. It really should be a required reference/text for every physics and engineering program in the country. All the mathematics is here, but the author also lucidly explains in words and diagrams the physical meaning of these three operations. Many students learn how to manipulate these operations, but they often have trouble understanding exactly what they mean. This book is easy to read (how many textbooks can you say that about?) and there are lots of problems to illustrate key points after each chapter. The author uses the mathematical formalism to solve some basic problems in electrostatics to provide real-world examples. After working through this book, you'll not only be able to calculate the curl of a vector function, you'll know exactly what it means.

If you've been introduced to paritial integration then you'll be able to follow this book. I wish I had know about it while I was taking multivarible calculus and electrodynamics. There are a number of solved problems at the end of each chapter. A great way to foster your intutive knowledge of those little electrons.

This book does it all. Without being overly verbose, it explains div, grad, and curl in concepts clearer and maybe even more precise than you would see in some upper level vector analysis classes. The integral definitions for grad and curl are given and well-justified. Homework problems and a number of examples from E & M give practice and application. It's an easy read, relatively cheap, and I think it's great supplementary material for any math or physics student.

It's been over two decades since I first studied vector calculus from my old textbook on electromagnetic fields and waves (Lorrain and Corson, Freeman, 1970). I really enjoyed that class, and remain fascinated by the beautiful mathematics involved in the classical field equations of electromagnetism. When I saw Schey's book on the shelf in Boulder, Co., I immediately picked it up and flipped through the pages. This wasn't the book I'd set out to find (I wanted a good book on Photonics, to commemorate the conference I was attending at NIST on fiber-optic measurements) but I decided it would be fun to read it as a refresher course.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

If you want to learn vector analysis, this is the book to get. It covers the basics of vector calculus, inlcuding surface integrals, the divergence and curl of vector fields and gradient operators, as well as Stokes and Green's Theorem. Unfortunately, there is no real material here on tensors, which would have been helpful, but for all of the hopelessly confused math, physics, and engineering students, this item is a godsend. I used it to teach myself the subject while my professors were busy confusing me. A very clear, lucid and amusing introduction. Should be required reading for aspiring engineers and physicists.