Vector Calculus

by Jerrold E. Marsden, Anthony Tromba

Now in its fifth edition, Vector Calculus helps students gain an intuitive and solid understanding of this important subject. The book's careful account is a contemporary balance between theory, application, and historical development, providing it's readers with an insight into how mathematics progresses and is in turn influenced by the natural world.

Genres: Mathematics, Textbooks, Science, Reference, Nonfiction, Calculus, Academic

704 pages, Hardcover

First published January 1, 1976

Reviews

[trivialchemy · Rating 5 out of 5]

This is probably the most thorough, clear, and beautiful mathematics text I've ever used. No, I won't sell it to you.

[Sharath Chandra]

I had seen a lot of bad reviews about this book by frustrated math students. I personally think, that this is the best book that explains concepts of curl and divergence clearly. All those who thinks that this book is rigorous must spend a little bit of their time in reading the first few pages where mathematical symbols are explained. These symbols should be applied seriously and all theorems and example problems should be reread until the concept is clear This is a must read for people trying to understand electromagnetic Theory.

[محمد · Rating 5 out of 5]

This is easily the most beautiful mathematical book I’ve ever read.

[Adam Lantos · Rating 4 out of 5]

While I intended to read all of it, after finishing with chapter 2, I found Colley's "Vector Calculus" to be much better than this. So, I will not be reading this book anymore.

I originally wrote
"I am going to provide a review of this book while I am going through it I will edit each time I go through one chapter.

On the whole book(I will edit this as I go on):
Everything is explained in a very clear way. Lot's of examples and problems and the answers to half of them are very useful. There also a lot of helpful illustrations. The book isn't rigorous in a way that would satisfy a mathematician, but for me-a physicist-it's ideal. It's as rigorous as a non-mathematician would like it to be. When there is no proof for something, the author provides motivation for it. That's great! Check the chapter-specific mini-reviews for stuff I did or did not like.

Chapter 1:
In the first chapter, I will give this a 4-star rating. This chapter is considered an easy one; an introduction. The are examples that make sure that you know the basics of every thing that the author tries to teach you. While everything in this chapter were clear(and the many illustrations helped a great deal in this), the author rarely goes the extra step to provide a deep insight. Having said that, I must also say that everything that the author tries to cover are as clear as it could be. Insights are gained through the MANY problems. Now, the problems are much more difficult than the examples; the examples are there to make sure that you got the main point of each subject. I like how the exercises go from very easy and gradually escalate to hard. There are some creative(good) exercises here, but if you take into consideration the large amount of problems you will conclude that there could/should be much more of them. There are problems that have to do with physics; nothing fancy, just straightforward stuff. I also encountered one or two problems(of the over 100 of the chapter) that failed in their effort to guide the reader to a solution, but I think every book that contains so many exercises has this problem. I enjoyed the historical notes(which many times contain biography of a great physicist or mathematician) but I won't take this into consideration while I am rating the book, because this is not the essence of it.

Now, for more section-specific things:
1) The chapter on spherical and cylindrical coordinates systems didn't satisfy me. There were some great problems but the examples were too simple in comparison. Also, the unit vectors in each coordinate system(for example the "azimuthal unit vector") were left to find as a part of two problems! I think the author should prove them because they are both important and a bit tricky to find. The overall chapter felt rushed with the author only giving the information in a raw manner. But, keep in mind that at the start of the chapter, the author warns the reader that he supposes the reader is familiar with those coordinates systems, so I don't hold the "plainness" of this chapter against him.
2) The problems on finding lengths and equations of planes or lines were pretty good.
3) Some things from linear algebra would be really helpful if they were put into a "review chapter" and connected with the rest of the chapter(for example linear independence, basis, etc. These might not be important to the understanding of the chapter material but they would provide further insight through connections). Some things are presented here, but in no solid way.

I analyzed various specific things I did not like with this chapter, but I insist on giving it a 4-star rating because being so clear and giving motivation behind everything is a rare thing for a book. I understand everything in this chapter and I did not have a hard time with anything. This means the book is very good.

Chapter 2:
I just started this and it seems like I am getting to the good stuff now. Mathematical notation is being used more-and that's good-and I think it might get a bit more rigorous than I initially thought. "

[Damián]

Me sirvió MUCHO para preparar Análisis II. Totalmente recomendado, se explica todo de una forma sencilla, pero que a la vez te permite llegar a algún lugar más profundo.

[Devastatingwildness]

Un manual muy útil para estudiantes de ciencias físicas e ingeniería. Pasé de apenas hojear el manual de de Burgos, Cálculo infinitesimal de varias variables, a usar este porque me parece mucho más ágil. Esto tiene sentido porque no es lo mismo estudiar una asignatura anual de cálculo de varias variables, como en matemáticas, a un cuatrimestre o menos dirigido a poder usar los teoremas clave en el día a día del estudiante de ciencias naturales. Aquí lo importante es entender los conceptos de forma operativa y con explicaciones claras, sin necesidades de demostraciones o excesivo tecnicismo, y ver los ejemplos útiles.
Me gustaría haberle dedicado más tiempo, ya que me parece divertido al estar muy dirigido a la física. E incluso tiene notas históricas lo cual para mí es fantástico (aunque a otros les parezca innecesario).

[Jake · Rating 4 out of 5]

I love the sections on curves and manifolds.

[Laura L. Van Dam · Rating 5 out of 5]

One of the best textbooks I used in university; clear and with lots of useful examples.
There is another volume of solved problems of this book that proved particularly useful to prepare practical exams.

[Spyros Thomopoulos · Rating 5 out of 5]

Clear and straightforward presentation of the theory and plenty of solved examples and quite difficult unsolved excersices

[Logan · Rating 2 out of 5]

The material is explained in an approachable manner. However, it comes at the expense of essentially covering the same material as third semester calculus while claiming that it is advanced. This book should be used for an advanced/honors calculus III, not an upper-level math class. The exercises have some infrequent errors, which can be annoying. Check your answers with a solutions pdf! In addition, the exercises are tedious and occasionally rely on integral tables! You will often skim the exercises to strengthen your conceptual understand, only to find exercises that test your ability to evaluate tedious integrals. The only reason I believe this book should be opened is for its introduction to differential forms. It's simple and well-written.

[jortegga · Rating 5 out of 5]

De forma no irónica el libro es brutal y precioso, es increíble lo muy bien conexo que está todo y lo claro que queda sin perder formalidad. El análisis el la parte que menos me gusta de las mates y me ha hecho apreciarlo mejor. Lo añado aquí porque es el libro que más tiempo y recursos me ha tomado y justo me lo acabo de terminar ahora.

[Mohamed Ahmed Aboushama · Rating 5 out of 5]

Wow wow wow ❕
The book is so fucking good and easy. thanks to all the people that helped this book be alive all these years...

[Tue Le · Rating 3 out of 5]

Challenging exercises but lousy explanations. Stewart's Calculus does a better job.

[isabela. · Rating 4 out of 5]

tbh this was nice i liked it

[Nelly3 · Rating 5 out of 5]

So good

[Anye · Rating 2 out of 5]

Trauma

[Crown SRSS · Rating 5 out of 5]

Read this book for calculus 3 (multivariable calculus). It was good.

[Anna Papagianni · Rating 5 out of 5]

Amazing textbook!

[Allie · Rating 1 out of 5]

The history of math sections are absolutely pointless. Who has time to read those extra sections? They just take up extra space in the book. It could probably be about half the size if they cut those sections out.



Surprisingly light, though, for the width of the book.

[Daniel · Rating 4 out of 5]

concise, good examples, no fucking around.

[Al Capwned · Rating 5 out of 5]

Thorough explanations and enough exercises, this is probably the best book about Vector Calculus.