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

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If you are an undergraduate engineering or science major, then you need to get a copy of this old classic and become good friends with it. If you are a graduate student or a professional in some branch of engineering or science, and you have not already read this book, then sneak out and get a copy before anybody finds out. (You can pretend that you really knew this stuff all along.) Seriously, this book should be considered Math 101 for scientists and engineers. You simply cannot get by without knowing the basics of vector calculus, curvilinear coordinates, Gauss' law, Stokes' theorem, and of course, the protagonists Divergence, Gradient, and Curl, known to their friends as Div, Grad, and Curl.

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!

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 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.

This text provides a systematic introduction to vector calculus in a very readable, informal format. Key concepts like divergence, curl, gradient, line integrals, surface integrals, Divergence Theorem, and Stokes Theorem are introduced in the context of investigating solutions to electrostatics problems without requiring the reader to be especially familiar with physics. I particularly enjoyed the humor that is woven into the text. ("Thus, the anguish of remembering the form of curl F in Cartesian coordinates can be replaced by the pain of remembering how to expand a three-by-three determinant.") I would highly recommend this concise book to students of physics, engineering, and mathematics. It is particularly suitable for self-instruction.

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.

I first checked this book out of a library, and was so pleased I decided to buy it. I am enrolled in a graduate level fluid mechanics class after being out of school for a few years and I needed to brush up on my vector calculus. This book was great for that job. It explains the concepts of divergence, gradient, curl, directional derivatives, line integrals, surface integrals, Stoke's Theorem, and Divergence Theorem with good mathematical rigor and notation, yet also with the "words between the lines" that most math texts leave out. In other words, accompanying each equation you will find a sentence or even a paragraph describing what exactly took place between steps.

Additionally, the author makes a point to describe the concepts behind the jargon and equations. When you took vector calculus the first time (if you ever did), could you explain in words what a "curl" is, or a "divergence"? This book attempts to do so, and does so fairly well (as well as one could given that these concepts don't have the easiest translation into words).

Furthermore, the author even has a sense of humor and made me laugh a few times. When was the last time you laughed out loud at a math book?

Finally, this book also includes applications to physics such as electrostatics (the recurring thematic problem of the book is Gauss's Law), fluid dynamics, and work.

Not only was it a great refresher, I wanted to own this clear and simple book as a reference.

I took multivariable caculus and vector calc not terribly long ago. While I did okay in the class, I finished it feeling two questions were still a bit hazy: 1) What exactly is the significance (physical and computational) of all of these techniques and operators (div, surface integrals, curl...etc.)?, and 2) What's the big picture...how does this all fit together?.

While this book is absolutely fantastic, from it's laid back writing style to its clear emphasis on applications, it is not a textbook in the traditional sense. In other words, this book makes a great supplement (without a doubt, the most commonly recommended one) and a fantastic review book, but it should not be read to learn vector calc for a first time. Its proofs are fiated and incomplete (the author is the first to admit this) and scope is limited (again, the author seems to take pride in this fact). But who cares; that's not the point. The point is this: vector calc is one of the most beautiful ways to mathematically model various important areas in science and mathematics, and Schey isn't going to finish with you until you have a really great INTUITIVE understanding of what it is you're actually doing.

Scientists, engineers, and math lovers unite! This book is fun, easy to read, and great for filling in the gaps. It's been a classic for three decades, and it's a mistake for anyone with an interest in vector calculus to pass it up.

As an undergraduate, I've already taken a course in multivariable calculus, and I found this text to be a quick and accessible review (you can read it in one sitting). That being said, I'm not sure if I would have gotten much out of it if I hadn't already had exposure to the subject. In addition to single-variable calculus, the book assumes knowledge of double and triple integration, which I didn't learn until I took a course in multivariable calculus. If you really want to learn the subject, but can't take a course in it for whatever reason, I'd recommend actually spending the $100 or so it takes to buy a textbook, and working through it on your own. I used "Vector Calculus" by Marsden and Tromba, which was adequate for my purposes. If you're not good at translating mathematical jargon into everyday language, you'll find "Div, Grad, and Curl" to be a useful supplement. I had better insight into the physical meaning of the del operator after reading it.

This little book is a real gem! For a long time till I read this I had been left confused and puzzled about the physical intuition behind these ubiquitous vector operations. [btw, even today I don't claim to be a god of Vector Calculus ;)] This little book is a very handy book to flip through in such a case to clear up some of the concepts you failed to grasp during your college lectures... It keeps a good balance between providing the intuition in the form of examples from Physics [Electric Field mostly], as well as pretty-rigorous math [If not 100% hard core rigorous] as well as the geometric insight that is so necesary to appreciate the usefulness of these concepts.
The fact that it is a small volume, and the light and easy going style of it's prose makes for great positives.
The problems given at the end of each chapter are also adequately challenging.
On the whole a very nice book. Highly recommended.

I used the first edition of this book during a vector calculus course at M.I.T. way back in 1980, and it gave me exactly what I needed. The book describes vector operations in a very straightforward manner, and gives an excellent sense of what Green's theorem, Stokes' theorm, and the Divergence theorem are all about. Since then, those sorts of vector calculus operations have never been a problem.

If you've taken (or are in the process of taking) vector calculus (whether an intro in multivariable calculus or as a class itself) this book is indispensible for support.

It's best feature is the fact that physics and engineering students can benefit from it's applied viewpoint, specifically on electric charge, potential. etc.

The title of the book is established quite well in that this book is a relatively light read and that the reader will be able to comprehend vector calculus with an understanding of why scientists use vector calc in the first place.

Overall, an excellent read with the answers to selected exercises placed in the back allow the reader to monitor learning.