1. May 24, 2012

### the_kid

I realize the term 'Advanced Calculus' is rather vague, so to be more specific I'm looking for a textbook covering multivariable analysis. I've taken a look at Shifrin's Multivariable Mathematics and Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms, and these seem like quite nice books. However, I'm concerned that they may be less than rigorous. What does everyone think of these books? In addition to these, what other (potentially more challenging) books are recommended?

2. May 24, 2012

### micromass

In order of less challenging to hard:

Analysis on manifolds - Munkres
Calculus on manifolds - Spivak
Multivariable real analysis I,II- Duistermaat, Kolk

3. May 24, 2012

### the_kid

I'm assuming Shifrin and Hubbard are significantly easier than those three books. I realize rigor is impossible to quantify, but, if you had to say, how much easier are they? Any thoughts on this would be appreciated!

4. May 24, 2012

### demonelite123

i haven't read the 3 books micromass has mentioned but i have used hubbard's book and it is definitely rigorous. he proves practically everything and leaves the more difficult proofs in the appendix. the text along with the proofs in the appendix can actually be used as an analysis text.

hubbard is usually a first year/second year math text. if you have already taken multivariable calculus in a reasonably rigorous fashion. That is if you covered things like the derivative as a linear mapping, pathological functions where equality of mixed partials fail, derivatives of matrix functions, implicit function theorem, integration over surfaces (hubbard treats integration over manifolds), integration theorems, etc. then you could probably move on to something like spivak's calculus on manifolds.

all the topics i listed above (and more) are covered in hubbard.

5. May 24, 2012

### transphenomen

I am almost finished with the Hubbard book and I am also almost done with baby rudin. They both touch the same topics, but Hubbard has pictures and examples whereas Rudin is really dense. For example, Hubbard has a whole chapter on linear algebra while Rudin just has a couple pages. They both go over the same proofs and so have the same rigor. The main difference is that Rudin is more general in using metric spaces while Hubbard uses R^n. All in all is took me two years to get through Hubbard; mostly because I got stuck on the Inverse Function Theorem at the end of chapter 2 and didn't want to go on until I got that down. I finally just said "to hell with it" and went on anyways and spent the last 6 months doing the rest of the book.

6. May 25, 2012

### theorem4.5.9

Last edited by a moderator: May 6, 2017
7. May 25, 2012

### alissca123

Last edited by a moderator: May 6, 2017
8. May 26, 2012

### Sankaku

9. May 26, 2012

### V_Permendur

the Arfken and Weber Mathematical Methods for Physicists should be rigorous enough for you - its the grad level text.

Or you could try a book titled "Advanced Engineering Mathematics" physics majors and engineers often take upper division math classes from a book with this such name. however i do not recommend the Kreysig or the Zill versions of that book.

My favorite book was called "Advanced Calculus" by Kaplan. It was my grandpas book, first edition, printed in like 1951.

10. May 26, 2012

### mathwonk

I believe Shifrin's book is completely rigorous. i.e. complete definitions and proofs.

A harder book, also completely rigorous, is Foundations of modern analysis, by Dieudonne'.

11. May 26, 2012

### deluks917

Hubbard to me does everything the hard way. Spivak Calc on manifolds does a similar amount in 1/4 as many pages.

12. Apr 11, 2013

### julypraise

I write this answer on 2013, so it is quite late but I still hope this be helpful.

I know a book that can definitely satisfy you.

It is Advanced Calculus by Fitzpatrick.

I've recently read Munkres's and Duistermaat's.

The former one is very very good. It is perfectly rigorous bot not only that it is just

beautiful and so comprehensible. I felt like drinking this book, very smooth.

The only down side was that I did not contain every topic that I looked for,

such as Lagrange Multiplier Method.

The latter one seems also good. But I must say reading this was very tough though I didn't

even read much of it. The selection of Lemmas and Theorems are not that efficient nor beautiful

nor organized; I actually feel that they are quite messed up. But this book is certainly rigorous

and contains strong Theorems on many many topics.

Now the book Advanced Calculus by Fitzpatrick is quite easy and very smooth to read.

It also contains basic topics such as Lagrange Multiplier Method.

Simply put, this is just basic and rigorous. Though the size of multivariable stuffs is not

much, it will lay down every base for you to go on to more advanced topics in this area.

So I think the best way is to read this and then read a more advanced book such as the

ones mentioned above.