# A dilemma between two multivariable calculus books

• Calculus
Hi,

I am a math major, and currently studying vector calculus. Since I am feeling that I don't really learn it properly, I am going to re-learn it again in the summer.
I would like to improve both my theoretical and computational skills. I am also searching for a book that starts from the beginning to deal with differential forms, and a book not full with physical interpretations since I am poor at physics.

My background is two courses of linear algebra, single variable calculus, and multivariable calculus of real-valued function(from the limits function of several variables up to triple integrals).

After a long search, I am considering(at the moment) two books:
1. Multivariable mathematics by Shifrin.
2. Vector calculus, by Hubbard.

But not sure which is preferred(or neither of them).
Unfortunately, both contain linear algebra chapters which are not necessary for me.

Thank you.

Sadly, I do not have access to Shifrin's text, but I know other of Shifrin's texts and they're all pretty good. So I doubt you can go wrong with it.

But anyway, I'm a big fan of Hubbard. Hubbard contains linear algebra, multivariable calculus in ##\mathbb{R}^n##, manifolds, some differential geometry and differential forms. Here are some of the strenghts of Hubbards book:
• Very beautiful and intuitive treatment of forms. It's the best treatment of forms and vector calculus I've seen pretty much in any book. Forms are a difficult subject to understand, but Hubbard makes it easy.
• Rigorous proofs of every result (possibly in the appendix)
• Cares about how to compute things and not only about how things are done in theory.
• Very intuitive and neat treatment of the Gauss curvature
• Takes derivatives of matrix operations, which is very useful later.
On the other hand, while the problems in Hubbard are nice, I feel that if you're interested in theory, you might want to supplement it with some more theoretical problems. I recommend Spivak's calculus on manifolds as a good source of extra theoretical problems. I do not recommend Spivak at all as a main text however.

mr.tea
Sadly, I do not have access to Shifrin's text, but I know other of Shifrin's texts and they're all pretty good. So I doubt you can go wrong with it.

But anyway, I'm a big fan of Hubbard. Hubbard contains linear algebra, multivariable calculus in ##\mathbb{R}^n##, manifolds, some differential geometry and differential forms. Here are some of the strenghts of Hubbards book:
• Very beautiful and intuitive treatment of forms. It's the best treatment of forms and vector calculus I've seen pretty much in any book. Forms are a difficult subject to understand, but Hubbard makes it easy.
• Rigorous proofs of every result (possibly in the appendix)
• Cares about how to compute things and not only about how things are done in theory.
• Very intuitive and neat treatment of the Gauss curvature
• Takes derivatives of matrix operations, which is very useful later.
On the other hand, while the problems in Hubbard are nice, I feel that if you're interested in theory, you might want to supplement it with some more theoretical problems. I recommend Spivak's calculus on manifolds as a good source of extra theoretical problems. I do not recommend Spivak at all as a main text however.

Thank you for the informative answer.

I'll start from your last paragraph. That's the reason I am considering another text. I have seen Spivak's text in our library. It doesn't have rigid structure(like Rudin's PMA for example), so it looks like you need to work in order to decipher a paragraph of text(there could be few definitions in a paragraph or two). But I used it to clarify some things. But since he doesn't provide computational problems, then as you said, it's not recommended to use it as the main text.

I have read some posts here, and it sounds that Hubbard is really a great book, although some pointed out about things that the authors tend to do rather unnecessary lengthy.

Can you point out what are the weaknesses of Hubbard?

Thank you again!

Can you point out what are the weaknesses of Hubbard?

So, I think
• Not enough theoretical (proofy) problems.
• Tends to take large detours. For example, it takes a long detour in probability theory, which you could easily skip. I don't personally find this problematic, but if you're looking for a book that will teach you everything quickly, then this is not good for you.
• A bit too wordy every now and then.

So, I think
• Not enough theoretical (proofy) problems.
• Tends to take large detours. For example, it takes a long detour in probability theory, which you could easily skip. I don't personally find this problematic, but if you're looking for a book that will teach you everything quickly, then this is not good for you.
• A bit too wordy every now and then.

I remember you recommended two books in multivariable calculus with different approach, one in differential forms and another one in geometric algebra. Which one is better suited for students going to advanced topology and analysis?

mr.tea
mathwonk
Homework Helper
2020 Award
Unfortunately I do not have access to Hubbard, but it is well regarded here, and I can certainly commend his mathematical genealogy. He is a student of Adrian Douady and has a doctor of science in France, a degree higher than a PhD, and has taught at Harvard and is now at Cornell.

I myself liked Spivak's Calc on Manifolds, but I knew the material to some extent before, and found it a nice succinct account of the most important results. I did not on the other hand like his proof of Stokes' Theorem, since it is so formal as to be unreadable. I understood that theorem by reading the short account of it in two variables in Lang where it immediately became clear (as I recall) that it is just Fubini, used to reduce to the one variable case, and then just the fundamental theorem.

There are basically, after developing differentiation and integration, three theorems there, inverse functions, fubini, and stokes, (oh also change of variables in integration), and he does a nice job on the first two of them as well as the basic properties of differentiation and integration. I myself also liked his careful development of differential forms algebra.

He also made me see clearly the role of partial derivatives, i.e. I understood that the derivative is a local linear approximation to a function, to within "little oh", but I did not appreciate that to actually calculate it and to know when it exists, you need partial derivatives. So that is what is missing in infinite dimensional banach spaces and why computing derivatives there, rather than just defining them, is hard. Many books claim that one might as well do differential calculus in banach space since it is the same definition as in finite dimensions, without honestly telling you this important difference.

i did once use spivak as a text in a course and it was a disaster, only one student being able to read it.

Unfortunately I do not have access to Hubbard, but it is well regarded here, and I can certainly commend his mathematical genealogy. He is a student of Adrian Douady and has a doctor of science in France, a degree higher than a PhD, and has taught at Harvard and is now at Cornell.

I myself liked Spivak's Calc on Manifolds, but I knew the material to some extent before, and found it a nice succinct account of the most important results. I did not on the other hand like his proof of Stokes' Theorem, since it is so formal as to be unreadable. I understood that theorem by reading the short account of it in two variables in Lang where it immediately became clear (as I recall) that it is just Fubini, used to reduce to the one variable case, and then just the fundamental theorem.

There are basically, after developing differentiation and integration, three theorems there, inverse functions, fubini, and stokes, (oh also change of variables in integration), and he does a nice job on the first two of them as well as the basic properties of differentiation and integration. I myself also liked his careful development of differential forms algebra.

He also made me see clearly the role of partial derivatives, i.e. I understood that the derivative is a local linear approximation to a function, to within "little oh", but I did not appreciate that to actually calculate it and to know when it exists, you need partial derivatives. So that is what is missing in infinite dimensional banach spaces and why computing derivatives there, rather than just defining them, is hard. Many books claim that one might as well do differential calculus in banach space since it is the same definition as in finite dimensions, without honestly telling you this important difference.

i did once use spivak as a text in a course and it was a disaster, only one student being able to read it.

Thank you for the information about Spivak's book.

I can't use only Spivak's book since it doesn't have a computational problems.

Thank you!

mathwonk
Homework Helper
2020 Award
I did have a copy of Shifrin's advanced calc book but no longer do. I liked it at the time. It was the textbook for his second year advanced honors calc course, incorporating linear algebra. His proof of the inverse function theorem as I recall is the standard fixed point contraction lemma proof using only "completeness" that works also in infinite dimensions. Spivak's proof is a special one that uses "compactness" of the ball, and hence only works in finite dimensions, but I rather liked it as offering a bit more geometric insight. When I taught advanced calc myself i developed proofs of several results from topology e.g. about non existence of vector fields and the fundamental theorem of algebra, using differential forms. I noticed that Shifrin also has all the same corollaries in his book that I had thought of, and I like it for that. I would agree with micromass that all Shifrin's books have a similar style, so if you like that style, say in his linear algebra, or abstract algebra boks, you should be happy with it here too. As I recall, he never overexplains, and writes for a fairly literate reader, and always has excellent problems. In his courses he is skillful at knowing what to include and what to take for granted. I would say someone like John Lee's books are perhaps more gentle. I rather liked the book by Wendell Fleming, but based on recommendations here, and the little I have found to read about it, I myself would like to have a closer look at Hubbard. On the other hand, if money matters to you, here is a copy of fleming for under $6. That's unbeatable. http://www.abebooks.com/servlet/SearchResults?an=fleming&sts=t&tn=functions+of+several+variables one very handy thing about fleming for me as a student, was that I had not learned the prerequisite course at all well, but Fleming had a 44 page appendix summarizing literally everything I needed to know from that course, before trying to read his book. Last edited: I did have a copy of Shifrin's advanced calc book but no longer do. I liked it at the time. It was the textbook for his second year advanced honors calc course, incorporating linear algebra. His proof of the inverse function theorem as I recall is the standard fixed point contraction lemma proof using only "completeness" that works also in infinite dimensions. Spivak's proof is a special one that uses "compactness" of the ball, and hence only works in finite dimensions, but I rather liked it as offering a bit more geometric insight. When I taught advanced calc myself i developed proofs of several results from topology e.g. about non existence of vector fields and the fundamental theorem of algebra, using differential forms. I noticed that Shifrin also has all the same corollaries in his book that I had thought of, and I like it for that. I would agree with micromass that all Shifrin's books have a similar style, so if you like that style, say in his linear algebra, or abstract algebra boks, you should be happy with it here too. As I recall, he never overexplains, and writes for a fairly literate reader, and always has excellent problems. In his courses he is skillful at knowing what to include and what to take for granted. I would say someone like John Lee's books are perhaps more gentle. I rather liked the book by Wendell Fleming, but based on recommendations here, and the little I have found to read about it, I myself would like to have a closer look at Hubbard. On the other hand, if money matters to you, here is a copy of fleming for under$6. That's unbeatable.

http://www.abebooks.com/servlet/SearchResults?an=fleming&sts=t&tn=functions+of+several+variables

one very handy thing about fleming for me as a student, was that I had not learned the prerequisite course at all well, but Fleming had a 44 page appendix summarizing literally everything I needed to know from that course, before trying to read his book.

Thank you for the information.
Is Fleming's text appropriate to a math major? self-study? how challenging are the exercises in the text?

I will try to search Shifrin's books in our library to see the style. According to your description it sounds a great book.

Thank you again!

mathwonk
Homework Helper
2020 Award
fleming's exercises look good too. a mix of routine and theoretical ones. it was the texbook when i took math 55 at harvard (the course famous now as "possibly the most difficult undergraduate course in the USA") from lynn loomis, before he brought out his own book with sternberg. i.e. yes, fleming is definitely aimed at math majors. of the three books used in math 55 while i was there, namely steenrod nickerson and spencer; fleming; loomis and sternberg, i personally think fleming is the best to learn from. some people said loomis was actually following dieudonne's foundations of modern analysis as his own guide, which is also excellent, but not at all as friendly for young learners as fleming.

what are the changes in 2nd edition of fleming?

mathwonk