Book Description

This book introduces students to vector analysis, a concise way of presenting certain kinds of equations and a natural aid for forming mental pictures of physical and geometrical ideas. Students of the physical sciences and of physics, mechanics, electromagnetic theory, aerodynamics and a number of other fields will find this a rewarding and practical treatment of vector analysis. Key points are made memorable with the hundreds of problems with step-by-step solutions, and many review questions with answers.

Reviews (6)

An outstanding tutorial reference
I love this book. I've owned three copies of it over the years and I can honestly say that I would not have achieved the final class of degree in Physics that I did without it.

The learning curve is very gentle - really nothing is assumed about the reader's background beyond basic integral and differential calculus. The concepts of vectors are introduced one by one, and the book builds logically towards its final stages (introductory tensor analysis) via, inter alia, dot and cross products, partial differential operators on vector spaces (grad, div, curl, Laplacian etc.), line and surface integrals (along with vital allied therorems such as Stokes' and Green's theorems), and general theory of curvilinear coordinate systems (in which the differential operators are refined and generalised).

This book is absolutely ideal for an undergraduate course in Physics, Electronic Engineering or Vector Analysis.

Best book on Vector Analysis...
Best book on Vector Analysis. No match for it. Must buy.

Good by itself
Although this series of books is intended to supplement a class textbook, this one is pretty good in helping you learn vectors by itself. It explains all the terminology and gives you quick examples.

For the problems here, there are some solved problems, which walk you through the process of finding the answer. The supplementary problems, to help you test your knowledge, have the answers there. This used to bother me because I wanted to see if I could get the answer. Here, the author is just trying to help you master the process. You can always cover up the answer.

Topics in this volume include tensor analysis, curvilinear coordinates, vector integration and differentiation, integral theorems, and dot and cross product. All are helpful and easy to understand.

If there is more than 5 stars?
The one published in 1959 deserves to be one of the finest books written about vectors .The way it deals with the subject prepare the reader smoothly in mastering the basics of vector analysis, its for the engineer, physicist and mathematician.

By the way the full name of the book is "Vector Analysis and an Introduction to Tensor Analysis"

Great Practise With All Basic Vector/Tensor Analysis
This is great as a preparatory or supporting text. I worked through virtually all of the 'supplementary' problems and found the chapters on curvilinear coordinates and tensor analysis very useful preparation for the study of General Relativity texts. Major parts of Landau and Lipschitz 'Classical Theory of Fields' and many other texts were readily accessible after doing the sums from Spiegel. Eminently suitable for independent study. ... Read more

Book Description

Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques, with large number of problems, from routine manipulative exercises to technically difficult assignments.
... Read more

Reviews (6)

Amazing, it's only eleven dollars but worth HUNDREDS
This is a book from which you can learn about tensors and really KNOW WHAT'S GOING ON!!! Sure, so you can complain about it's slight lack of rigor--big deal!!!!! Once you're done reading this book, move on to the books that DO have rigor (if you're a mathematician-type rather than a physicist-type), but if you want an introduction to the theory of tensors which provides true intuitive understanding of what tensors are, why they are useful, and how the idea of tensors arose, then buy this book. This book requires almost no prerequisites except for a good background in vectors, matrices, and certain aspects of multivariable calculus. Buy it NOW, and thank God that it is published by Dover.

not so so rigorous
Lots of material for the price, but this is one of those maths book with a "physical approach", IMO. Definitions for instances, like the definition of a Tensor, aren't always enounced clearly.

This just make things look more complex and different than what they are for no gain.

I believe that the book "Tensor calculus on manifold", same editor, Goldberg/Bishop does a better job: more rigorous and more concise.

Rigorous, yet informal enough to be a lot of fun.
Many years ago, this became the first book I had ever read about tensor calculus, differential geometry, or classical field theories, and I still have not found a better treatment of any of the three subjects anywhere else. I'm now not very far from a PhD in General Relativity theory, and I very rarely need to use any mathematics which I didn't first learn as a freshman undergraduate while reading this book independently. I owe a great debt to Lovelock and Rund, and could not recommend this book any more highly than I am right now.

One of the best books ever
I don't know how they did it but, this is the book you want to buy if you're trying to learn differential geometry, especially if you're learning general relativity. It takes you from the concepts you are already familiar with into differential geometry faster than any other book I've ever tried (and I've tried many!). Before you know it, you are comfortable with covariant derivatives and Lie derivatives and.. well the list could go on. Do not be turned off by the reputation of Dover books-- "cheap and not worth it!" This is a gem.

For those of you learning GR: Buy this book and Schutz's "Geometrical Methods of Mathematical Physics." Read Lovelock and Rund first and then dive into Schutz's book. This will provide you with the necessary mathematical background to handle Wald's "General Relativity" with (some amount of) ease. You might want to try Schutz's "A First Course in General Relativity" before Wald's more advanced book.

I've read many glowing reviews on Amazon about books that I "must have" and, quite frankly, they turned out to be poor choices. But in this case I have to say you "must have" this book! It is that good. And it's cheap, so if you do not agree with me, it's not much money out of your pocket.

THe Mathematics of General Relativity
The authors present a thorough development of TENSOR CALCULUS, from basic principals, such as ordinary three dimensional vector space. Tensors are generalizations of vectors to any number of dimensions (vectors are type (1,0) tensors, diff. forms are type (0,1) tensors). One of the key principles of General Relativity is that if physical laws are expressed in tensor form, then they are independent of local coordinate systems, and valid everywhere.

Chap. 1: Preliminary Obs.-- Chap. 2: Affine Tensor Algebra in Euclidean Geometry-- Chap. 3: Tensor Analysis on Manifolds -- Chap. 4: Additional Topics from the Tensor Calculus -- Chap. 5: The Calculus of Differential Forms -- Chap. 6: Invariant Problems in the Calculus of Variations -- Chap. 7: Riemannian Geometry -- Chap. 8: Invariant Var. Principles and Phys. Field Theories -

Chap. 8 covers a good deal of General Relativity. This book is a worthy addition to any mathematical library. ... Read more

Reviews (23)

The Perfect Book for Vector Calculus
After you've learned calculus from a book like Stewart's, this is the definite next step. It includes lots of proofs for most of the theorems. I love the way the textbook is organized. And all the explanations are incredibly clear. The choice of notation makes all the formulas a lot clearer than in other books. Although I am not as big a fan of math as I am of physics, I must say that I really enjoyed reading this book a lot, and believe you will too. This book seriously makes advanced calculus incredibly fun!

Better proofs and clearer explanations are needed
When I was a second-year math student, I found that "Vector Calculus" needed to provide students with clearer and more rigorous explanations and proofs of theorems. Too many statements were expected to be belived on face value. Specifically, I found no intuitive justification for the Jacobian Matrix anywhere in the book, nor any hint of a proof or explanation of its origins. Students will take to this already-challenging subject more if theorems are rigorously proved. This way they feel they're being treated like mathematicians instead of idiots. Moreover, the subject of multivariate calculus would feel like mathematics instead of a course in vocabulary memorization.

Not a good intro.
While some of my peers deem "Vector Calculus" to be a fine integration of theory and practice, I'd have to COMPLETELY disagree. From a teaching stand point, it is one of the worst texts out there (at least for a first course). At my university, some of the instructors have tried to use it as the text for the second half of a four quarter calculus sequence. This attempt has met with terrible failure, in my opinion. Most of my students (math majors and engineering students) found the book difficult and perplexing with few examples that pertained to the material they were required to learn. Luckily, the professor for my course was very good at conveying the ideas present without alluding to the text; nevertheless, I spent countless hours in discussion helping my students understand material that most standard texts would have clearly elucidated for them. In fact, at numerous points, the text becomes so involved with its own pedagogy that it neglects to delinate between important, must-know theorems and simply interesting facts.

In addition, only the very first exercises in a given section are useful for most students. A number of the later questions become interesting problems in some upper div. class, but have no bearing on the course at hand. Quite a few of them are not difficult but require "tricks" which often discourage the students by giving them the impression that they don't get the material simply because they couldn't come up with the solutions to these extraneous questions.

I would strongly recommend Stewart's text (for those of you on the West Coast) and Salas and Hille's text (for those of you in the Southwest).

Prehaps, Marsden's text would be o.k. for a more advanced course on vector calc. or as a go-between supplement for a more rigorous text.

Relatively weak as a standard textbook on vector calculus
I am well aware of the usefulness of these reviews in determining the applicability of a book for self-study; so let me address this quickly. This has got to be the worst vector calculus book available if you're looking to study the subject on your own!!! This book is frustrating and dry; please consider other self-study options!

Unfortunately, most people who use this text are required to for a class, and for whatever reason, this book has become somewhat of a standard at many universities. I used this book a while back in a Vector Calculus class at UT Austin, and I was largely disappointed by its contents.

First of all, the author of the book is dry and completely uninspiring. That's not to say that people read calculus books like novels, but the author presents the material from a strictly technical and theoretical perspective. Further adding to its blandness, the author (or the publisher) has opted for the cost-effective choice of using no color in the book. The graphs and figures are confused and lacking - often difficult to understand.

Now, the obvious rebuttal to my accusations will come from purists (hardcore math majors). I am, myself, a math (and physics) major, and though I am not saying that this text is completely inaccessible, I have to say that the author wrote this book wholly without imagination or sincerity. There is no emphasis on vector calculus' usefulness to applied mathematical sciences or other areas of math (if I do recall, though, a bit is addressed in association with integral theorems).

The only reason I give this book two stars is that the later parts of the book offer a peak at more advanced topics in geometry.

Last, and perhaps most inexcusable, the book requires an errata as a full supplement (I'm not exaggerating). This book is littered with errors, and not just grammatical typos! I suffered a couple of times on assignments due to incorrect formulas in the book. For example, the edition of the book I used gave the incorrect formula for the second derivative test! Now come on, they're actually charging people for this!!!

Worst Calculus book!!!
This is the most horrid book I have ever seen on the subject of Caclulus. Having taught from this book this semester, I can say it has over-simple examples, but NEVER EVER seems to give examples of how to work problems beyond the first one or two(and sometimes not even that). Really, if you use this book, you are doing your students a considerable disservice, and should be fired. ... Read more

Book Description

Reviews (12)

new visual metaphors
New visual metaphors for different kinds of vectors in 3-space: arrows, stacks, thumbtacks, and sheaves (corresponding to contravariant, covariant, and two forms of tensor). Visual and helpful proofs of Gauss's theorem, and Stoke's theorem, and div, grad and curl. I suspect the book would have been better had he included tensors explicitly. Very valuable for anyone doing vector analysis.

A Short, Fascinating Book. Buy This One.
I hardly need to add another Highly Recommended to the list of reviews. However, Professor Weinreich has assembled from his lectures an exceptionally interesting and intriguing geometrical approach to vectors. Not the conventional directed line segment approach, but one which questions which geometrical relationships are topologically invariant and which are not. This is not a difficult book, but I suspect that the more familiar the reader is with vector concepts, the more surprised and appreciative he will be.

Invaluable -- Div, Grad Curl ++
I actually have a few complaints about this book, but the core material is so helpful and instructive that they don't much matter.

This book explains vector and the beginnings of tensor analysis with new visual metaphors for vectors: lines, sheaves, thumbtacks, stacks. The dot and cross products can be visualized with these metaphors, and the various forms of Stokes/Gauss theorems proven visually.

This is great stuff for anyone going beyond the basics in vector analysis -- which would be anyone in pure math or physics, and some engineers.

You do need to use this as an adjuct to a conventional text or course.

This is the more sophisticated and general version of "Div, Grad, Curl and All That".

A great companion to math and physics
This book is deep! While lacking the formal rigor of vector analysis or exterior calculus this book attempts to remedy the lack of intuition that often accompanies such treatments (read the preface of the book).

In this book the author sneaks in clifford algebra, forms and applications to physics, he gives us a method of calculation that opens up the vector calculus you already knew and gives a great way to 'draw' many phenomenon in physics.

The author has an important agenda in this volume and that is to distinguish between objects that naturally behave differently. It has been the legacy of Gibbs and Heaviside for us to flounder in the 3-d application/misapplication of Hamiliton's quaternions. The reader is led to realize that identifying everything with contravariant vectors (arrows) is wrong and damaging to our intuition of phenomenon.

I highly recommend this book. It may seem hokey at first with odd names like thumbtack and swarm but it portrays deep mathematics in a beautiful manner. Work hard on it, apply it to physics and mathematics and be surprised at what you find! This sort of geometrical analysis is hard to find (try Gravitation by MTW or Applied Differential Geometry by Burke) at this level.

Remember it is meant to be an affordable companion to courses on vector and tensor analysis, and what a companion it is!

A Great Book too bad the tensor one never got here
This book is one of the best books on vectors that I have ever read, and believe me I have read many. It was one of the easiest reads that I have ever encountered in a math book. Everything is explained very well but a very interested student will have to go somewhere else to find meaning to many of the vector types. I would have like to to see more detail on N dimensions or even four since that is what relativity requires. Overall though a great book and an extremely easy read. ... Read more

Reviews (12)

The best introduction to Vector Calculus ever written
The author has written a carefully thought out introduction to the subject whose only assumptions are that you know the most rudimentary coordinate geometry and single variable calculus. From this all the classical subjects in vector calculus are built up using geometric ideas to motivate the definitions of the concepts. Typically the first course in vector calculus tries to get to Stokes Theorem and so on as quickly as possible without explaining what motivated these ideas. Much of the technical apparatus in vector calculus was used in modelling fluid dynamic flows in the nineteenth century, this is where the idea of "vector field" came from. As far as I know, this is the first vector calculus book I've read that defines a vector field, and next to it shows a picture of water flowing out of an upturned cup, with velocity vectors pointing in all directions. Just one picture captures the essence of the definition and immediately renders concrete something very abstract. There are many other examples in the book where a picture is shown of an abstract concept, making the definitions and theorems intuitive.
However, this book is not just pretty pictures, the calculus is built up in a rigorous manner (as far as a first introduction to the subject goes) and by the end of the book you are well placed to read your first book on manifolds and differential geometry. The book is not cheap, but if you think about it in terms of if you wanted to replicate this book you'd need at least 3 other standard textbooks, then its reasonable. Even advanced mathematicians would be surprised how much they could learn by looking at some of the pictures ! This book would be ideal as an appetiser before a main course of graduate differential geometry.

A solid, thorough treatment of multivariable calculus.
I used Susan Colley's Vector Calculus when I took multivariable calculus in the spring of '99. The book is very well written and I would definitely recommend it to anyone, but most especially to those who have a strong interest in the subject and aren't just fulfilling a requirement. Here is why--

When the reader is presented with an mathematical idea, it is nice to know where that idea comes from, and to be given whatever explanations or proofs are needed. An example of where Colley does this is in the chapter on the chain rule in several variables. This is a difficult chapter and Colley does an excellent job of explaining the underlying concepts (with lots of visual aids) where a less thorough author might have simply offered formulas and methods to solve a few specific types of problems.

Also, Colley introduces vector notation which, although at first unfamiliar, ultimately leads to a better understanding of the relationships between functions of different numbers of variables. For example, instead of the notation f(x,y,z,w,...) we have f(x->) (the arrow indicates that x is a vector). This notation, as well as the extensive use of matrices is very helpful and eliminates much confusion.

The visuals are simple and easy to understand, and the problems are appropriately designed, with plenty of very simple exercises for dealing with basic calculations, as well as very challenging and thought-provoking problems which require plenty of thought and help develop good mathematical intuition and visualization.

Overall this is a very good book, and it appears to me that the other reviews on this page come from neither a good knowledge of the book nor multivariable calculus.

Awesome
Professor Colley's book excels in all the areas one would look for including abundant examples, fine graphics, excellent graded problems, clear writing, good organization and so on. It stands out particularly for the author's sensitive presentation which not only presents the material in a clear, logical form but in such a way as to anticipate the questions of the reader. The use of geometric intuition is especially effective. Not being a great talent at mathematics, I found that this book clarified many ideas that I had not understood before. How the negative critics came up with their ideas is a mystery.

DO NOT BUY THIS TEXTBOOK!
This is most likely the most miserable math textbook I have ever had the misfortune to read. I showed it to my Mom (who has a Master's Degree in Math) thinking perhaps I was simply really stupid and she agreed that this textbook is abysmal. For example, the proofs often go into great detail concerning the arithmetic of the proof, but then the author fails to state clearly by what principle she jumps from one step of the proof to another. The great bonding session for the 13 members of my class has been comiserating over the poor quality of the textbook and trading tips about better Multivariable/Vector Calculus texts to buy in order to elucidate the material presented in Colley. Perhaps math professors find this book wonderful, but if the point of the text is to teach Multivariable Calculus to less than math-genius students (who are nevertheless somewhat adept at math) the text completely fails. If you have to get this text for your class because your teacher doesn't care whether his students actually learn the material, get an accompanying text to help. I have been using my brother's Multivariable text (by Larson, Hostetler, and Edwards) and another person in my class highly recommended Stewart's treatment of Multivariable Calculus.

A very poor mathematics text indeed
After having been taught Vector Calculus with this book last semester, I feel that I can say, without a doubt, it is terrible.

Although there are a plethora of problems with this text, I will only say its biggest: the poor examples the text tries to make use of which leave the reader feeling more confused than when he/she set out.

While this may (or may not) be a fun read for Calculus professors, it certainly should never be used to teach students the finer points of advanced calculus, and any professor who willingly uses this text over and over again should be fired. ... Read more

Book Description

This lucid introduction for undergraduates and graduates proves fundamental for pactitioners of theoretical physics and certain areas of engineering, like aerodynamics and fluid mechanics, and exteremely valuable for mathematicians. This study guide teaches all the basics and efective problem-solving skills too.

Reviews (8)

As an Outline...
This is not a book to learn tensor calculus from. It is an outline only, no greath depth or insight is presented. This book works perfectly as a supplement to a course in tensor calculus, or as a quick reference for the various techniques and concepts involved, provided one is already somewhat familiar with the material. It would be possible to learn the basics of tensor calculus from this book with some effort, and reflection on the implications of the concepts dealt with, however as a complete course in the subject it is insufficient, and I believe intentionally so.
The more modern aspects of tensor analysis on manifolds are largely ignored in this treatment, but also intentionally so, an approach which I found useful practically.
The book does not aim to be an all-inclusive course in the applications of tensor concepts to all areas of mathmatics, but rather a quick-reference guide supplementing more complete treatments, and as such, is largely successful.

Disappointed.
This book is substandard for the Schaum's Outline Series. The descriptions for the techniques are much too brief, and as a result, it's hard to follow what's going on. The summaries are so thin that it's even difficult to learn how to do the mechanics of tensor operations, a real deficit for an outline book! I have a pretty good background in advanced math, but I don't think I could learn tensor analysis from this book. I was especially disappointed because I have had good luck with other books in the Schaum's series. I'm planning on looking for a more traditional book with more discussion and background of the different techniques.

disappointing
I am surprised by the high marks given by other reviewers. The book has no insight, depth, or explanation about what tensor calculus is. It is merely a cookbook for doing some manipulations with summations over more than one index. You cannot learn tensor calculus from this shallow excuse for a book. If you want to learn tensor calculus and other advanced mathematics for physicists, pick up "Mathematics of Classical and Quantum Mechanics," by Byron and Fuller; it is the best math methods book I have ever used.

I'm Rating this A THOUSAND STARS
This Study Guide functions properly. If right now, you are reading the title "Tensor Calculus" and wish you understood it someday. But have absolutely no idea what a tesnor of rank zero is. GET THIS BOOK

My main goal was to understand General Relativity. But as you know, the mathematics of General Relativity is nothing but Tensor Calculus. I was particularly intrigued by the mysteries of the Riemann Curvature Tensor. The key to General Relativity. As soon as I purchased this book, I started studying Chapter 8, the "Riemannian Curvature" not knowing anything about the previous chapters. Hopeless I eventually turned to chapter 1 and gradually climed up the ladder. Then came my Golden Times in Tensor Calculus. I cracked the mysteries of the Riemann Curvature Tensor and at last I turned to General Relativity. I'm currently studying black holes thanks to this book.

Best Place To Start
This is probably the clearest ontoduction to tensor analysis that is currently on the market. It makes a quite difficult and messy subject seem pretty straightforward. It's best to know your vector calc in and out before attempting this book, but it's a godsend compared to some of the other texts out on the market today. A great guide for engineering and physics students and the price can't be beat. ... Read more

Reviews (7)

The older editions are better
I have the third edition and the latest edition of this book. In my opinion, the later editions have been dumbed down considerably. In addition to developing the basics of vector analysis, the older editions included splendid discussions of curvilinear coordinates, tensor analysis and touched on differential forms (the wedge product being introduced). While the latest edition features appendices covering specific applications such as Maxwells and Navier-Stokes equations, the explanations are far too brief to be of much use. I suggest that interested parties attempt to get used older editions whose content were presumably under the control of Davis (not Snider) being much better written and probably less expensive. The older edition had few typos with very useful and entertaining problem sets. A better alternative book might be "Vectors and Tensors in Engineering and Physics" by Donald A. Danielson in paperback.

Good for reference, not so good by itself
This book was used in my vector calculus class at UCSD. I think if I was in one of my upper division engineering classes it would be great to use as a reference to help me figure out problems. However, as a math book alone, this was not too helpful. Many of the explanations lack clarity and are from a mathmatician's point of view (in other words, very difficult for lower division students who are not math majors to understand). Also, a lot of the problems are practically impossible. The only reason I did alright in this class was because I had a great teacher. I would recommend this book for people who are in science or engineering classes and need a superficial introduction to vector calculus, but NOT for math classes that go into deep analysis of vectors.

Great book for scientists!
This book provides a great reference for people studying the physical sciences. I'm a Ph.D. student in Physical Chemistry and have been using it extensively to help out with some other courses. It might not be as good for someone with a more puritanical or abstract interest, e.g. a mathematician, but it's great for anyone that needs a handy reference that explains the rudiments of vector analysis.

This book is awful
This has to be one of the worst math books on the market. While it starts off OK (with the review of vector addition and algebra), the treatment of later material is terrible. The authors provide skimpy and disorganized explanations with insanely difficult problems. Get a good teacher for this class (or a better book) or you'll be in for a rough ride.

good introduction
I just finished a class on Vector Analysis that used this book. This book is good for those who are interested in vectors. The authors start off with the basics and then move quickly to the more difficult lessons. Basically this text treats the material the same as a calculus text does with a couple of exceptions. First, the book moves more quickly than a calculus text, so if you are shaky on the first part i would advise having a calculus book there with you. Second, this book shows the proofs in a more precise manner that the calculus text that I used (Stewart). You study the same equations and theories, but now you have harder exercises to do at the end of the chapter (with most of the answers in the back of the book). So I would recommend this book to those interested in vectors who are familiar with them. Otherwise you may want to look at a calculus book to get up to speed first. ... Read more

Book Description

This is an all-encompassing and exhaustive exposition of the theory of infinite-dimensional Unitary Representations of Locally Compact Groups and its generalization to representations of Banach algebras. The presentation is detailed, accessible, and self-contained (except for some elementary knowledge in algebra, topology, and abstract measure theory). In the later chapters the reader is brought to the frontiers of present-day knowledge in the area of Mackey normal subgroup analysisand its generalization to the context of Banach *-Algebraic Bundles. ... Read more

Book Description

Tensor Calculus and Analytical Dynamics provides a concise, comprehensive, and readable introduction to classical tensor calculus - in both holonomic and nonholonomic coordinates - as well as to its principal applications to the Lagrangean dynamics of discrete systems under positional or velocity constraints. The thrust of the book focuses on formal structure and basic geometrical/physical ideas underlying most general equations of motion of mechanical systems under linear velocity constraints.Written for the theoretically minded engineer, Tensor Calculus and Analytical Dynamics contains uniquely accessbile treatments of such intricate topics as:otensor calculus in nonholonomic variablesoPfaffian nonholonomic constraintsorelated integrability theory of FrobeniusThe book enables readers to move quickly and confidently in any particular geometry-based area of theoretical or applied mechanics in either classical or modern form. ... Read more

Reviews (1)

The definitive book on tensors in analytical mechanics
This book is not a text book. It is, in some sense, the final word on tensor formalism in finite degree of freedom (analytical) mechanics. It is one of the most scholarly books I have come across. The list of references is very exhaustive and the author is well read in the literature on the subject, not just in english, but also in russian, french, and german. The style is clear and concise, the notation is carefully chosen and summarized in a useful section where conventions, notation, and basic formulae are listed. ... Read more

Book Description

Proceeds from general to special, including chapters on vector analysis on manifolds and integration theory.
... Read more

Reviews (7)

The pros and cons of Bishop & Goldberg
I will briefly list the pros and cons of this book.
The pros are (a.) its price, (b.) the amount of material it manages to cover, (c.) it is quite complete - everything is formulated and proven within the text rigorously, and it covers a lot of ground (manifolds, tensors, differentiation and integration on manifolds, connections and Riemmanian manifolds) (d.) it does not require much background - nothing more than point-set topology and calculus. It even develops all the linear algebra it needs in a single chapter - quite admirable. (e.) the exercises are nice and instructive. (f.) It makes a good reference and supplement. (g.) It has a special chapter on Riemannian manifolds - quite good for relativity courses.
Now for the cons. (a.) the notation is a bit outdated. (b.) it does not treat infinite dimensional or complex manifolds. (c.) It sometimes leaves certain results for the reader to verify, which might annoy readers who simply want to get to a certain result as quick as possible. (d.) It is a bit dry. (e.) It lacks in concrete examples - that is not to say it doesn't have any examples, just that more would be much better, (f.) and this is chiefly aimed at physicits - it does not really focus on calculating things, which is what physics is all about, at the end.
Having said that, I honestly say that one can learn all about basic differential geometry from this book. I don't think seeing manifolds in R^n is a basic prerequisite for studying abstract diff. geometry. This book would be a good place to start - despite its age it manages to remain very relevant today. Finally, the reader is assured that the authors won't pull off any "dirty tricks" (since this is basically a mathematical book) - it's very important for the reader to be able to trust the book he's reading. And the price is fantastic!

A perfect starting point
This is a great book for an introduction to differential geometry. The only real prerequisite is calculus and some topology, making this book accessible to undergraduate students interested in Mathematics or Physics. The book covers a wide variety of topics and there are plenty of examples and exercises.
I guess the two reasons why I don't give this book five stars are (1)the notation in not entirely modern and (2) I have not managed to effectively use it as a primary textbook but as a supplument to a textbook.
It is certainly a great value for the price.

"Did you say relativity?" Find all the prerequisites here
This books is the perfect introduction to modern differential geometry, especially for people with a specific purpose in mind such as the study of relativity or analytical mechanics. This book is a very straight forward read. But that dosent mean it compromises on quality on the depth of the material presented. The exercises are great, as they illustrate the concepts just learned very nicely. One section leads very nicely to the other. As for the topology needed to study differentiable manifolds, it is developed in the beginnning, though its not the best "quick untro to topology" Ive seen. Of course you can skip some of the sections such as Paracompactness. The only consequence is that you might not be able to follow some of the proofs later on. The only other complaint is that in the few exercises on special relativity, they use the old "ict" coordinate system. Try to remember that this sysytem is frowned upon these days. But all in all an excellent read. And especially for the price you can buy this at.

Terse, not the best available
This is the unofficial "standard reference" on the subject. Lots of more advanced books cite it, and it certainly covers a lot of ground. The problem is it is a bit too terse. I say that as a math type. Were I a physics type I'd want a more visual approach. If you are learning tensor geometry though this book is worth having as a reference, and the price is right.

Well-written text
This is a terse treatment of differential geometry. It is perhaps too sophisticated to serve as an introduction to modern differential geometry. The beginner probably needs to see examples of two dimensional surfaces embedded in Euclidean 3-space and to do calculations with reference to such surfaces. For example,the use of coordinate patches to cover the 2-sphere. And then seeing how the change of coordinates in overlapping patches affects geometric objects such as vectors, 1-forms, and the metric tensor. This provides some grounding for the abstract treatment of manifolds and the tensors defined on them. Also a leisurely introduction to the geometry of curved surfaces, either classically, using the first and second fundamental forms, or the modern way, using the shape operator (which is equivalent). This motivates the more abstract treatment of connections, which become necessary when there is no underlying space to embed the surface in (Euclidean 3-space provides a notion of

connection (i.e. covariant derivative) that is geometrically clear; we have to axiomatize this notion when there's no natural space to embed in).

Though the book may not be suitable as a first text, it can be used in conjunction with a more elementary text. Alternatively, it could be used for a graduate course. Though there are now a plethora of other good treatments around, this book remains one of the classics,and furthermore its price makes it particularly appealing. ... Read more

Reviews (20)

Nonrigorous yet thorough explanation of vector calculus
This book is excellent for understanding the basics of vector calculus: divergence, curl, gradient, Stokes' Theorem, Divergence Theorem, surface integrals, etc. Most vector calculus texts gloss over the big picture of vector calculus with mathematical rigor that is usually too abstract for non-math students, preventing many from truly understanding vector calculus. Schey's book keeps the math to a minimum and makes sure the basic concepts of vector calculus are thoroughly, yet concisely, explained. The book is also nice to have as a future reference tool.

Best Intro to Vector Analysis
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.

A must for engineering and science students.
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!

Vector calculus presented from an applied approach
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.

Excellent book to brush up your Vector Calculus
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. ... Read more

Book Description

The first large-scale study of the development of vectorial systems, awarded a special prize for excellence in 1992 from France’s prestigious Jean Scott Foundation. Traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers created by Hamilton and Grassmann to the final acceptance around 1910 of the modern system of vector analysis. Concentrates on vector addition and subtraction, the forms of vector multiplication, vector division (in those systems where it occurs), and the specification of vector types. 1985 corrected edition of 1967 original.
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Reviews (2)

Thoughtful, Detailed History of Vector Analysis
How were the concepts of vector analysis developed? How did modern vector notation become widely accepted? Who were the key players and why did quaternions fail to gain acceptance? This book is extensively documented, scholarly in its approach, sometimes a bit slow, but overall it is a fascinating look at these specific questions as well as the fundamental issue of what factors promote or delay acceptance of revolutionary ideas in science and mathematics.

I did not become immediately engaged with Crowe's style and even set the book aside after reading the prefaces and first chapter. A few months later I returned to chapter two (in part due to a previous reviewer's high rating). And what a surprise - I suddenly found myself intrigued with Crowe's discussion of Sir William Hamilton's single minded focus on quaternions, the perseverance and genius of Hermann Grassmann, the critical roles played by Peter Tait and James Maxwell, and the pragmatic way in which Josiah Gibbs and Oliver Heaviside independently extracted key vectorial concepts from Hamiliton-Tait's quaternion analysis.

Crowe's book was originally published in 1967 by University of Notre Dame, Dover reprinted it in 1985, Crowe recieved the Jean Scott Prize by the Maison des Sciences de l'Homme (Paris)in 1992, and Dover reprinted it again in 1992. Dover should be commended for making such reprints readily available at affordable prices.

The discussion of Hamilton's quaternions does not require familiarity with quaternions, but some prior acquaintance might be helpful. I encountered quaternions in another Dover reprint: Matrices and Transformations by Pettofrezzo. Section 2-3 introduces quaternion notation, simple manipulations, and shows that addition and multiplication of quaternions is isomorphic with two particular sets of matrices.

Has quaternion analysis survived? See Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality by Jack Kuipers, Aprill, 1999. The reviews by readers are all five stars.

Thorough, intelligent and impeccably unprejudiced.
This book is a model of science history. Crowe manages to give one a clear view of the trends and moods of the time (1840-1900), the personalities of the various figures involved (Esp. Hamilton, Grassman and Gibbs), without sacrificing his most significant asset: the facts. ... Read more

Reviews (5)

Hard to follow
It's an old fashioned text, confusing and hard to follow.

okay book
this book dosen't take things from basics but goes to do high level calculus.

A good solid introduction
Synge and Schild is a good solid introduction to tensor calculus, as it is used by most physicists, and was used throughout the 20th century.

Antiquated
"The introduction of numbers as coordinates...is an act of violence..." -- H. Weyl.

If that's so, this is a very violent book. While it's true that physicists, particularly those working in General Relativity, were slow to abandon the coordinate approach, there can be little doubt that the sea of indicies form of Tensor Calculus runs counter to the modern approach to Differential Geometry, with its emphasis on abstract spaces, manifolds, bundles, exterior algebra, differential forms, diffeomorphisms, Lie groups, etc.

Physicists trained prior to the trend towards employing modern mathematics will likely be right at home with this book, which presents the tensor calculus in the form developed by Levi-Civita and Ricci in the late 19th/early 20th Century. On the other hand, classically trained Physicists tend to be hopelessly confused when confronted by modern Differential Geometry, which relies on so much more of the modern machinery from areas such as Topology, Global Analysis, and Group Theory/Representation Theory.

Students would be better served to pursue the subject framed in a more modern context. That means learning about manifolds and analysis on manifolds. The best introduction is probably Spivak's "Calculus on Manifolds", followed by Munkres "Analysis on Manifolds". Darling's "Differential Forms and Connections" and Sternberg's "Lectures on Differential Geometry" are well regarded, as is do Carmo's "Differential Geometry of Curves and Surfaces". A working knowledge of multivariable calculus, linear algebra, and elementary analysis are required for making heads or tails out of these books, even though they are introductory in nature. Having digested all that, one can now embark on the study of Riemannian geometry, say through do Carmo's "Riemannian Geometry", or Spivak's "A Comprehensive Course in Differential Geometry" (5 vols.). If you survived that then attentively study Kobayashi/Nomizu "Foundations of Differential Geometry" (2 vols., the diffeomorphism/bundle perspective) or Helgason "Differential Geometry, Lie Groups, and Symmetric Spaces" (from the perspective of Representation Theory) and go write your dissertation. Then come back and explain it all to me.

The best classical introduction to tensors
This is probably the clearest classical treatment of tensors you can find. Tensors are objects whose components transform in some linear and homogeneous way. This is the original definition, by Ricci, the founder of the theory. Today one prefers to define them as the members of some vector space and avoid talking of components. However, most physicists adhere to the classical formulation. After all this was the tensor calculus known to Einstein! Anyway the job is extremely well done: you end up knowing about parallel transportation and covariant derivative, curvature tensor and several applications. You'll be able to write the Laplacian operator in any corrdinate system whatsoever, and so on. I think the chapter on Integration is much more difficult than the others, but, then, invariant integration is the realm of exterior differential forms, and building them from tensors is inevitably clumsy. ... Read more

Book Description

Concise undergraduate text focuses on problem solving, rather than elaborate proofs. The first three chapters present the basics of matrices, including addition, multiplication, and division. In later chapters the author introduces vectors and shows how to use vectors and matrices to solve systems of linear equations. He also looks into special matrices—including complex numbers, quaternion matrices, and matrices with complex entries—transpose matrices; the trace of a matrix; the cross product of matrices; eigenvalues and eigenvectors; and infinite series of matrices. Exercises at the end of each section give students further practice. Prerequisites include algebra and (in the later chapters) solid geometry. 1961 edition. 20 black-and-white illustrations.
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Reviews (1)

Buyer Beware Mistakes & lack of Exercise Solutions
Just a quick review to warn people like me who buy books like these to help us in our autodidact pursuits. I bought this Dover book because I trusted Dover publications and am now disappointed to say that this trust is now in question. The book has some merit: the flow and examples are clear and well laid out however, THERE ARE NO SOLUTIONS TO THE EXERCISES!
Be aware that there are some glaring mistakes even to the beginner's eyes. For example, the author says "We may determine a 4X4 matrix..." and then represents this 4x4 with a 5x5 matrix. And he contradicts a commutivity theorem by saying that where two matrices A and zero share the same nxn row & column dimensions An+0n=A and then right after this he says A+0=0 on pp. 12-13. This is wrong! How am I supposed to trust further examples from this book? Sorry, I'll have to permanently shelve it...
Good luck.
IndiAndy
Aspiring Autodidact ... Read more

Book Description

No calculus needed, but this is not an elementary book. Introduces vectors, algebraic notation and basic ideas, vector algebra and scalars. Covers areas of parallelograms, triple products, moments, angular velocity, areas and vectorial addition, more concludes with discussion of tensors. 386 exercises.
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Reviews (1)

Vector Victorious...
With the name Hoffmann Banesh, this book need no review. He is the author of many wonderful mathematics books which include illustration. As the old saying "a picture worth a thousand words", this book use picture to solve the mysterious of vector which is so complex. It will help you gain a better understanding of vector if you're taking Physics or Vector Analysis class. ... Read more