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

by Harry M. Schey

This new fourth edition of the acclaimed and bestselling Div, Grad, Curl, and All That has been carefully revised and now includes updated notations and seven new example exercises. Since the publication of the First Edition over thirty years ago, Div, Grad, Curl, and All That has been widely renowned for its clear and concise coverage of vector calculus, helping science and engineering students gain a thorough understanding of gradient, curl, and Laplacian operators without required knowledge of advanced mathematics.

Genres: Mathematics, Physics, Textbooks, Science, Nonfiction, Reference, Technical

176 pages, Paperback

First published January 1, 1973

Reviews

[Dmitri · Rating 5 out of 5]

This math book is incredibly fun to read: clear, clever, and, believe it or not, gripping. It makes me wonder why people bother searching for enlightenment in ancient tomes. All one needs is a semester of calc, and this book.

[Adam Lantos · Rating 4 out of 5]

This book is all about intuition rather than rigor.
Pros:
1) Intuition: Non-rigorous derivations that are so intuitive that it will be easy for everyone to reproduce them and actually really understand the basic ideas behind them.
2) Applications to Electromagnetism.
3) Figures: A lot of figures mean that you better understand what the author says.
4) Quick read due to its small size.

Cons:
1) Not enough examples: Although the author has justified the existence of the 4th edition largely due to the insertion of more examples, I still think that they are not enough. And the reason is that most of them are trivial. At many occasions I have found that non-trivial exercises give small details that can largely enforce one's intuition and understanding. These kind of examples are simply not here. Maybe they are left as exercises, but it wouldn't hurt if some where examples.
2) Author's stubbornness to preserve the small size of the book: While the small size makes for a quick and easy read, it certainly would not hurt if the next edition was bumped up to 200 pages. A lot of meaningful things could be added as there are a lot of topics that beg for this kind of basic yet highly intuitive and pictorial exposition.
3) Applications ONLY to electromagnetism: Many students build their basic intuition about field lines on fluids, so it would be very reasonable to have some examples/applications to fluid dynamics. As a second point, I will say that applications to other subjects would have been a pleasant and welcome change from the many applications to electromagnetism.

Conclusion:
I give it a 4-star rating because while it has many very good points that differentiate it from other books on Vector Calculus, it also has some restrictions. These restrictions are just enough to justify a 4-star rating, although I would gladly give it an easy 5-star rating if it had just a bit more examples.

[Erickson · Rating 5 out of 5]

Very intuitive and concise. Good start before entering full fledged mathematical formulation of electromagnetism.

It also helps build intuitions for calculus courses which are more rigorous, since the author specifies in what sense he is being "sloppy" by stating what are needed to make some arguments work in rigorous manner.

[Nicholas Martinez · Rating 5 out of 5]

I recommend this book as a review of vector calculus rather than an introduction. I picked it up because I felt this material was skimmed over in a multivariable calculus course, and I was glad to find a brief yet thorough guide. The author relies on the reader to already grasp the mathematics, but he prefers application to formalism. Many of the explanations begin with an application and what the concept of interest means in that context; in fact, the whole book is sort of a story of how mathematicians learned to calculate electromagnetic field magnitudes, because measuring it directly was impractical. As a student new to the subject, I was suggestible to Schey's implications throughout the text that it may not be possible, only to have him report the simple, elegant solution.

There are many practice problems available, but I was not really committed to attempting these problems. Perhaps they would have elevated my understanding. However, I am fresh out of multivariable calculus, so I have already had some practice with these types of problems. I had mainly wanted some conceptual depth when I started reading.

Overall, it packs a punch for its brevity and price. Highly recommended, especially for students.

[Aneece · Rating 4 out of 5]

Ideal for the physics student who wants to develop their intuition for the basic theorems of vector calculus. A proof may reassure you that a theorem is true, but give you no insight into how it was discovered. I've read that mathematicians prefer proofs, physicists derivations. This book satisfies the desire for derivation and motivation over rigor.

[Andrew · Rating 4 out of 5]

This is a great, accessible primer on vector calculus. The exercises represent a good range of difficulty, and the informal style of writing makes the concepts easy to grasp in the absence of classroom instruction.

[Frank · Rating 5 out of 5]

I really dig it. I recommend the Physics undergrad read this the summer before their first year of university studies. It will prepare you well for.

[Ja Se · Rating 5 out of 5]

This book does what it sets out to do, providing intuition and a sense for the tools of vector calculus and tying it into how it has been explored in real life. It lacks rigor and formalism, just as advertised. My understanding of the subject has gotten considerably better, especially after going through all the problems in the book (which I would venture to say are not optional at all). It's apparent that the author had a vision for the book in its succinctness and what it's trying to achieve, and I would say he has manifested it very well.

A critique, perhaps, not of the book itself but of notation commonplace in physics vs mathematics, is that it is an absolutely arduous task to look up anything about curvilinear coordinates if your memory for multivariate calculus isn't perfect due to different conventions.

[Andrew Davis · Rating 4 out of 5]

A quite interesting refresher of essential concepts in vector calculus in preparation for electromagnetic theory. A quite detailed exposition of the topic with a great deal of examples and exercises to follow. The discussion covers topics not only in the cartesian coordinates, but also cylindrical and spherical ones.

[Kiira · Rating 4 out of 5]

my life is finally easy now.

[Mostafa Alkady · Rating 5 out of 5]

An extremely useful book for vector calculus that you'll probably go back to more than once especially if you're into Physics. Very easy to grasp, provided with figures, and links most of the topics to their applications in real life. Readers just need to know some basic calculus and a glimpse of multivariable calculus to understand it all. I highly recommend it.

[Ronald Lett · Rating 4 out of 5]

I used this book while studying physics in high school. If you are a physics student struggling to put precise physical intuition into divergence, gradient and curl operations, this text is a quick study.

[Morgan · Rating 5 out of 5]

This was a great overview of vector calculus, as well as an interesting investigation into how to find the electric field. It's well written, at times funny, and easy to follow.

[Chad · Rating 5 out of 5]

I read this book in a day, it's really quite accessible given you have some basic background knowledge of calculus. I checked this book out for my continued project to develop a historical roleplaying game for a physics class on electricity and magnetism. I have been reading a lot of primary source documents from some of the field's founders-- Ampere, Faraday, Maxwell-- and it is HARD reading. You have multiple difficulties: first, these are written from an outdated scientific paradigm, so there are terms and concepts with which I am unfamiliar. And second, because I am not an expert in the current paradigm, I can't even "translate" from one paradigm to another. It's hard going. This is my attempt to familiarize myself with electricity and magnetism as taught in the twenty-first century. I have taken physics classes in undergrad, so I have a very basic understanding. I also have my engineering background-- so all the math minus the electricity and magnetism concepts. This book brought it all together, essentially developing Maxwell's equations step by step introducing the math as needed. From the title, I was hoping to be able to more intuitively describe what the divergence, gradient and curl are. I have used these before as a chemical engineering graduate student, but I have to say I was never comfortable understanding them intuitively. They were just symbols. I also am uncomfortable with vector notation in general, always preferring to translate them to cartesian coordinates. Perhaps because that's what you ultimately "solve" these equations in. I wish I had this book in undergrad, it would have made things make a lot more sense.

[Lucille Nguyen · Rating 3 out of 5]

I read this book because I heard raving reviews from others in the sciences and physics talk about it. Having some knowledge of the subject already, I figured it would help me gel my intuition. And it was decidedly disappointing.

First of all, starting with surface integral is an... interesting choice. Probably not a good one, from an organization standpoint. Applying the math to applications in physics, particularly in electrodynamics and the heat equation is good for seeing applications. However, spending time introducing the subject and going through a problem before showing the definition or the formalisms does not work well for me. And ending, instead of starting, on the gradient (which is intuitive for anyone who's done basic derivatives) seems like a poor choice. Many books start with the gradient for obvious reasons.

Perhaps this book is good for newcomers to the subject, who have more knowledge of the sciences than they do formal mathematics. For me, I found little to love about the format, and thought that the author prioritized conciseness over intuition. Maybe to not scare off scientists who might be intimidated by large math books, Schey kept it short, personally as a devourer of large math books I am a fan of the more pages the merrier. Different strokes for different folks I suppose, I have known many who have loved this book.

[Kasparas · Rating 3 out of 5]

This book attempts to explain the main ideas behind the differential calculus while avoiding going into too much detail behind the mathematical derivations. And it does a pretty good job at it. The concepts are well explained, derivations are easy to follow, and most importantly the text is concise. It’s easy to pick up this book and read it in a day or two - perfect for a student. There are also a lot of complementary exercises which is helpful.

The book attempted to directly apply the derived mathematical ideas to describe the laws of electromagnetism, but in my opinion this could’ve been done better. Likewise, I don’t think that the explanation of div, grad or curl operators was the best one I’ve seen. It was a new one to me, maybe even original, but not the best. I think an emphasis on the graphical representation would’ve made the concepts clearer.

Overall, not the best book I’ve read but still great for a student in a hurry.

[Aryan Prasad · Rating 3 out of 5]

An informal introduction to some basic concepts in Vector Calculus. Quoting the book:

The word "Formal" in this context is a euphemism for "useless"

All concepts are motivated with the help of electrodynamics, and as such will be great for students of physics. The material is often times not so rigorous and a lot of hand waving is involved in proofs, but to be fair the author admits that he is being "sloppy."

For people more interested in he concepts rather than the applications, they should supplement this with a more standard book (Thomas Calculus, Advanced Calculus or any other standard text may do)

[Ariful Islam · Rating 5 out of 5]

A short and concise book for getting intuition on vector calculus as a tool for physics. It starts out by explicitly setting a goal to find the electric field due to any distribution of charges, and then it approaches the goal step by step, introducing the concepts of vector calculus to overcome the obstacles one by one, in a very logical progression. Along the way you also learn theorems that are not so closely tied to the goal. So overall its an intuitive first introduction to the subject that requires elementary calculus as the only prerequisite. Highly recommended for a first time exposure to the subject.

[Jeremy M-K · Rating 3 out of 5]

This was suggested reading in college; I earned a bachelor’s of science in applied mathematics. Later in life, I wanted to study my maths again, and so I read this informal text on vector calculus. (Spoilers follow) My college instructor, Prof Embid suggested the book but warned us the solutions are incorrect. He was right. I knocked down to three stars because of the errors. At least in a few cases the errors are nonsensical enough that a keen mathematical intuition can see they are nonsensical. Unfortunately if you want to know which solutions are right and which are wrong, you just have to try to work ‘‘em out yourselves. Challenge accepted?

[Chayan · Rating 3 out of 5]

Overall content wise it's good, but didn't like the organisation and presentation.

For anybody willing to learn or revise vector calculus, I would recommend the first chapter of Introduction to Electrodynamics. In just 55 pages, Griffiths covers everything in this book and more, builds very strong intuition, covers applications and presents all that in a much better, learner-friendly way.

[Nattapon Chotsisuparat · Rating 5 out of 5]

An informal text on vector calculus called "Div, Grad, Curl, and All That" was first released in 1965. This textbook only has roughly 150 pages. You will learn the vector calculus used in electrostatics from this book. It is simple to follow and comprehend. This is a smart pick if you want to understand the fundamentals of electrostatics and vectec calculus.

[Lucy Low · Rating 5 out of 5]

good book for basics

[sine · Rating 5 out of 5]

A short overview of vector calculus, teaching the basics with enough problems to master the material.

[Jari Havela · Rating 2 out of 5]

A slight disappointment

[Mac n cheese · Rating 2 out of 5]

Not a good resource

[Jenny Nguyen · Rating 3 out of 5]

Not too keen on vector calculus

[Selva · Rating 4 out of 5]

Beautiful book on vector calculus, explained things in an elegant manner, must read books for math graduates and engineers.

[Cliff · Rating 5 out of 5]

It amazes me that I graduated from Colby College with a BS in math without knowing Stokes' theorem, or anything significant about differential equations!

[Ashleyanstaett · Rating 5 out of 5]

I would highly recommend anyone taking a course in vector calculus OR E&M to pick up this little gem. I didn't discover it until the end of my course in vector calculus, and I wish I had used it as a companion piece for the course. The presentation of surface integrals, div, grad and curl in my vector calculus course were fairly abstract and difficult for me to grasp so Schey's use of E&M to illustrate these concepts makes them a bit easier to digest. It's also nice to get a glimpse of the utility of these important theorems. This book is definitely no substitute for more rigorous proofs and derivations, but it would be a great refresher, and is a perfect companion piece for quite a few subjects. I also found the practice problems to be really helpful, as they are not overly-difficult and tend to reinforce the most important concepts.

[Andrew · Rating 5 out of 5]

I ran across the Vector Function derivatives in the title in a physics book. This book explained the mysteries hinted at in the other book. Limited to three dimensions, limited in its rigor, but within these limits crystal-clear about the how and why of Div, Grad, Curl and the equations in which these derivatives occur.

Short and sweet.