Looking for a rigorous multivariable calculus book

[Adgorn]

Hello everyone.
I'm about to take Calc 3 next semester and am looking for a rigorous book to work with on multivariable calculus. I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution manual. At first I considered "Calculus on manifolds" but from what I've been told it's too dense and will be better appreciated as a second exposure. I've also considered Courant & John Vol-II but having red the beginning of Vol-I, I dislike the approach the authors take, and the explanations feel somewhat more hand-wavy to me (at least compared to Spivak).

So is there any book on the subject that will be in the same vein as Spivak (and preferably with ?
Thanks to all the helpers.

 

[BvU]

There's an insight on this :
https://www.physicsforums.com/insights/self-study-calculus/

But isn't it more sensible to take the book your Calc 3 will use ?
(Do you have a syllabus showing the topics that will be treated?)

Otherwise there are

Stewart
Varberg, Purcell, Rigdon

 

[Infrared]

At first I considered "Calculus on manifolds" but from what I've been told it's too dense and will be better appreciated as a second exposure.

Perhaps you should take a glance at the book and see if it looks readable to you? I think it should be a reasonable follow-up to his single variable calculus book.
There's also a book of Hubbard that I've seen highly recommended on this forum (but I haven't looked at it myself).

 

[PeroK]

Hello everyone.
I'm about to take Calc 3 next semester and am looking for a rigorous book to work with on multivariable calculus. I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution manual. At first I considered "Calculus on manifolds" but from what I've been told it's too dense and will be better appreciated as a second exposure. I've also considered Courant & John Vol-II but having red the beginning of Vol-I, I dislike the approach the authors take, and the explanations feel somewhat more hand-wavy to me (at least compared to Spivak).

So is there any book on the subject that will be in the same vein as Spivak (and preferably with ?
Thanks to all the helpers.

Just a thought, but there may be a limited market for fully rigorous multi-variable calculus. Rigorous single-variable calculus is essential for pure mathematics, of course, but after than you are more likely to move on to other pure mathematical disciplines, such as algebra, complex analysis, functional analysis and linear algebra etc.

Multi-variable calculus is generally a tool for applied maths and physics, where there is less concern with rigorously proving everything. And, to some extent, the proofs are extensions of the single-variable cases in any case - like the multi-variable Taylor series.

As an aside, I watched a brilliant set of lectures on mathematical physics by Carl Bender (they are on YouTube) and he stressed how much of the work in his field has not been rigorously proven - and that looking for rigorous proofs is a major drawback and ultimately a limiting factor to what you can achieve. That said, there are some amazing results that have been rigorously proven.

 

[Frimus]

Give a try to "Elementary Real Analysis" by Thomson and Bruckners.
It is rigorous and includes a multivariable calculus too. You will realize that in spite that you can get it free from classicalrealanalysis.com, it is by no means *cheap* and hand waving book. It is intended for undergraduates planning their career in mathematics.

 

[jasonRF]

I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution manual.

By "gone through" I'm assuming that means you worked through the proofs yourself and solved a reasonable fraction of the homework problems. Yes?

If so, then one option worth a look is "vector calculus, linear algebra and differential forms" by Hubbard and Hubbard. It was written for the honors sophomore-level math at Cornell. I don't know Spivak so cannot say if it is in the same vein, but the parts I have read were very well written and quite rigorous. It is a very long book (perhaps 800 pages?) and I'm not sure that is what you are looking for.

jason

 

[MidgetDwarf]

By "gone through" I'm assuming that means you worked through the proofs yourself and solved a reasonable fraction of the homework problems. Yes?

If so, then one option worth a look is "vector calculus, linear algebra and differential forms" by Hubbard and Hubbard. It was written for the honors sophomore-level math at Cornell. I don't know Spivak so cannot say if it is in the same vein, but the parts I have read were very well written and quite rigorous. It is a very long book (perhaps 800 pages?) and I'm not sure that is what you are looking for.

jason

There is a major leap going from something like Spivak Calculus to Spivak Calculus On Manifolds. Moreover, he asked for solution manual + rigorous multi-calculus book. To be frank. Thus us an oxymoron. How would one want rigor, and yet also ask for solutions manual? It defeats the purpose.

I myself recently took a class using that book, and found it difficult, although the teacher was no help and utterly uses. This was having things like Complex Analysis, Intro Analysis, and Topology under my belt. Let alone the other proof based math classes. Although one can definitely learn from it, I don't think many will be able to absorb any meaningful mathematics. The OP requiring a solutions manual greatly supports this.

The problem with rigorous multivariable calculus books is that they generally use the language of linear algebra and introduces concepts of topology (closed/open sets, boundary/interior etc). Moreover, it is much different than the standard Cal 3 course (usually aimed at engineers/physics and other science majors).

The Hubbard book is good. Not as rigorous as say Spivak, but I gained more from that book than Calculus on Manifolds. Serge Lang was also clear. I may just read Spivak or go with Munkres Analysis.

 

[mathwonk]

I recommend taking a look at the classic books by Apostol, for your purpose probably Calculus , vol. 2. At a more advanced level, you might look at his Mathematical Analysis. These books are as rigorous as Spivak, maybe even more scholarly, but possibly less "fun".

Another excellent choice is Functions of several variables, by Wendell Fleming.

Another one I have learned from was originally titled Analysis I, by Serge Lang, but has changed names since then. It includes one and several variables in one volume. (Lang's second volume, Analysis II, perhaps now called Real Analysis, is an advanced graduate level text that I do not recommend for undergrads or even beginning grads.)

 

[Adgorn]

But isn't it more sensible to take the book your Calc 3 will use ?
(Do you have a syllabus showing the topics that will be treated?)

Courses in my Uni rarely have a single book that they follow, at most they feature a list of recommended books with a stern warning that the materials of the books may not overlap the course material, and the books features are usually not as comprehensive as I want them to be.

The syllabus is about what you would expect: Euclidian N-space, basic topology, functions between euclidian spaces, Taylor's formula and extremums of functions, Lagrange multipliers, multiple integrals (standard and improper), line integrals, conservative fields, vector analysis, gradient, curl, divergence, surfance integrals, Green's, Stokes' and Gauss's theorems and a few more things.

Just a thought, but there may be a limited market for fully rigorous multi-variable calculus. Rigorous single-variable calculus is essential for pure mathematics, of course, but after than you are more likely to move on to other pure mathematical disciplines, such as algebra, complex analysis, functional analysis and linear algebra etc.

Practically, you're probably right that over-emphasis on rigor might bring more harm than benefit and perhaps even occlude certain aspects, however I've come to realize that I'm simply more comfortable with a subject (and its problems) when I know it to its fullest detail. It's a matter of personal preference more than anything else. I will however take a peak at these lectures.

By "gone through" I'm assuming that means you worked through the proofs yourself and solved a reasonable fraction of the homework problems. Yes?

All of them, in fact. T'was quite a journey.

If so, then one option worth a look is "vector calculus, linear algebra and differential forms" by Hubbard and Hubbard.

Yes, I've seen it recommended several times. It's indeed quite a tome but looks rather appealing at the same time. I'll have to decide If I have time to go through it during a semester but it's a worthy contender from what I've seen so far.

 

[Adgorn]

There is a major leap going from something like Spivak Calculus to Spivak Calculus On Manifolds. Moreover, he asked for solution manual + rigorous multi-calculus book. To be frank. Thus us an oxymoron. How would one want rigor, and yet also ask for solutions manual? It defeats the purpose.

I disagree with the notion that rigor somehow negates the utility of a solution manual. Any student is fundamentally limited in their ability to review their own solutions.

First, even if one believes that one solved a problem in the most rigorous way possible, mistakes and misconceptions are unavoidable. Perhaps the student misinterpreted the conditions for the validity of a certain theorem or applied it in some wrong manner. Having a list of solutions handy is helpful in rectifying such misconceptions and allows the student to solve other problems correctly, while the lack of such external assessment may solidify the mistake and harm the understanding of the material as the student repeats it unchecked

Second, in many of the more computational problems, which tend to contain dozens of sub-sections, it is simply more convenient to have the solution a page flip away than to insert each and every one into a computer to ensure its validity (or to go over every single algebraic operation several times).

Third, most problems tend to have several solutions which may differ substantially. A solution manual may expose the student to methods and ways of thinking that he would not have conceived on his own, as I can attest happened to me many times in my experience with Spivak.

Lastly, dismaying as it may be, sometimes the student simply cannot find the solution on his own, as time is a limited resource and in some cases so is sanity. Therefore in such cases checking a solution is the more pedagogically preferable option, as it would again entail exposing the student to a new method that he could not come up on his own (provided of course one does not give into the temptation of checking too early, which I consider myself capable of doing). In these cases a solution written specifically for the problem, more often than not by a person of considerable experience in written explanations, is much more convenient than detailing the entire problem, its background and the student's background in a forum such as this one.

The problem with rigorous multivariable calculus books is that they generally use the language of linear algebra and introduces concepts of topology (closed/open sets, boundary/interior etc)

In that case I ought to be grand, as a I believe I have sufficient experience in both. I will however look into Munkre's, as I heard it is an expanded version of Calculus on Manifolds.

I recommend taking a look at the classic books by Apostol, for your purpose probably Calculus , vol. 2. At a more advanced level, you might look at his Mathematical Analysis. These books are as rigorous as Spivak, maybe even more scholarly, but possibly less "fun".

I'll certainly look into the book as I heard it is among the best, although it appears a rather small portion of the book is actually dedicated to multivariable calculus, with the rest concerning algebra or differential equation (which I'm already familiar with). Does the book cover all the material that's expected from a (possible advanced) course on the subject?Thanks to everyone for all the wonderful suggestions, I didn't except such an embarrassment of choices. I'm sure I'll find one of these that will suit my fancy :).

 

[mathwonk]

As you may know, differential calculus is the science of approximating non linear mappings by linear mappings. In one variable, linear mappings are just multiplication by scalars, so there is no learning curve. In several variables the subject is called linear algebra. Hence Apostol begins his volume on several variable calculus with a series of chapters on linear algebra in several variables, and their application to diff eq.

After these chapters, he treats several variable calculus in detail. Thus although you are correct that his book contains more than several variable calculus, it contains only that as well as needed prerequisites. So I repeat my recommendation of that book as well suited for your needs. I.e. it contains possibly more than you asked for, rather than less.

I don't know your career intentions, but I am myself a retired professional research mathematician, and this of course influences my recommendation. Good luck with your future.

 

[jasonRF]

Hi Adgorn,

I just wanted to point out that I would trust mathwonk's recommendation more than mine. Hopefully you have access to a library to look at the different options, although that might not be very likely these days...jason

 

[Adgorn]

Alright, so after boiling it down to Apostol Vol 2 vs Hubbard & Hubbard I've decided to go with the latter. Having gone through previews for both books and their table of contents it seems Hubbard will surely cover everything I'll come across in my course as well as some nice bonus material. Also, it seems as rigorous as I would like and the friendly prose of the book indicates that it will be a fun read.

Thanks again to everyone, you've helped me a lot.