Spivak's level vector calculus book

[carlosbgois]

Besides Apostol 2, is there any good, rigorous and suited for self study book on this subject? Thanks

 

[Petek]

I assume that the title of your post means that you're aware of Spivak's Calculus on Manifolds, but are looking for other texts. Here are three you might consider:

1. Analysis On Manifolds by Munkres. Covers topics similar to Spivak, but in a more leisurely fashion.
2. Advanced Calculus: A Differential Forms Approach by Edwards. Again, similar to Spivak, except introduces differential forms right at the beginning.
3. Functions of Several Variables by Fleming. More advanced than the other texts (uses Lebesgue integrals), but is still intended for undergrads.

You can search inside all three texts at Amazon to get a better idea. IMHO, some of the reviews on Amazon are not very helpful. If I had to choose one for self-study, I'd go with Munkres.

 

[xristy]

Courant and John - 2 volume Introduction to Calculus and Analysis is quite good. Years ago I followed Spivak Calculus with Volume 2 of Courant and John at university so I have some fondness for it.

 

[deluks917]

Loomis and Sternberg is a very nice free book. Its quite challenging though.

 

[intwo]

I assume that the title of your post means that you're aware of Spivak's Calculus on Manifolds, but are looking for other texts. Here are three you might consider:

1. Analysis On Manifolds by Munkres. Covers topics similar to Spivak, but in a more leisurely fashion.
2. Advanced Calculus: A Differential Forms Approach by Edwards. Again, similar to Spivak, except introduces differential forms right at the beginning.
3. Functions of Several Variables by Fleming. More advanced than the other texts (uses Lebesgue integrals), but is still intended for undergrads.

You can search inside all three texts at Amazon to get a better idea. IMHO, some of the reviews on Amazon are not very helpful. If I had to choose one for self-study, I'd go with Munkres.

I would recommend taking linear algebra and maybe even some real analysis before attempting Spivak, Munkres, or Edwards.

Unfortunately, I don't think that there is a multivariable calculus textbook in the same style as Spivak's Calculus. You might like Vector Calculus by Marsden and Tromba. It's in a style that is more rigorous than Stewart Calculus, but not as rigorous as Spivak's Calculus. There will be a new version sometime this month that is supposed to contain the typical Definition, Theorem, Corollary layout.

 

[Petek]

I thought the OP was referring to Spivak's Calculus on Manifolds, but it's more likely he meant Spivak's Calculus. If so, then my suggestions are inappropriate. Fellow posters: Spivak, Apostol, Rudin, et. al. authored more than one book. You'll get better replies if you're more specific.

 

[carlosbgois]

Thanks for all who helped, and sorry about my bad question. I'll explain it more specifically:

I've finished a first course on single variable calculus, and I intend now to review some parts using Spivak's Calculus (not calculus on manifolds), which I found to be a very nice read and far better than the books I used.

As in the next semester I'll be starting multi-variable calculus (calculus B/2), I'm looking for a good textbook, not so easy as stewart's, but not so hard that I take hours on each page.

Petek: I'm aware of Spivak's Calculus on Manifolds, but does it cover a standart Calculus B course? It seems so small to fit the whole subject of most other books I've seen, like stewart's 2, or marsden vector calculus.

Sorry if it isn't very clear again.

[Edit]By the way I know basic linear algebra: maps, inner product spaces, adjoint, orthonormal operators, self-adjoints, basically what's covered in Lang's linear algebra (though not as deep as it's taught there because I used an easier book for a start), and I have nothing on analysis.

 

[deluks917]

Spivak is a little light on computations unless you do most of the exercises. Calc on mainfold is good if you are willing to do the problems. The other books in this thread are also good choices.

One thing I have learned is that larger books often contain less information than shorter ones. Certainly Spivak has more content then a book like Stewart.

 

[Petek]

Spivak's Calculus on Manifolds is at the level of an upper division math course, probably best used after taking a course in analysis of one-variable real functions. The level is about the same as the other books in my first post. So, it's more advanced than a typical course in multivariable calculus.

 

[alissca123]

Functions of Several Variables - Wendell Fleming is my favorite...

I also like Advanced Calculus of Several Variables by C. H. Edwards

 

[sponsoredwalk]

Two books:

Advanced Calculus - Nickerson, Spencer, Steenrod.
http://store.doverpublications.com/0486480909.html

Review by C.B. Allendoerfer

In 1957, Nickerson, Spencer, and
Steenrod wrote a new advanced calculus textbook which was in effect an
introduction to the techniques of modern analysis. This book bore little
resemblance to the existing texts in the subject, and was not successful in
replacing them. However, it made the others obsolete; every text written
since then must reckon with the Nickerson-Spencer-Steenrod conception
of advanced calculus.
link

Am dying to get this book.

Calculus of Several Variables - Goffman
This book is almost identical to Spivak's book, except for the fact that the explanations
are far clearer but I would definitely read both Spivak & Goffman. If anybody reads either
of these books (Steenrod/Goffman) please post your experience with them, would be nice to read.

 

[mathwonk]

Opinions of course differ, but if you don't already have it, I suggest you don't bother with Nickerson, Spencer and Steenrod. I have never met or heard of anyone learning the subject from this book. Loomis and Sternberg is only a little less useless. These books are mainly for showing off, by the authors that is. Sort of "look how abstract and difficult I can make an important subject look!"

A book that many people in the past considered one of the best is the vector calculus book by Williamson, Crowell and Trotter.

That link about Spencer et al...from Hugo Rossi's introduction to his own calculus book, which seems never to have caught on, actually tells you how unsuitable the Nickerson book is. I.e. Hugo says there that he taught from it to a class of "exceptionally brilliant" students at Princeton and found it necessary to essentially write another different book to fill out the treatment in Nickerson et al.

I myself have a copy of Spencer et al, since Lynn Loomis required it for his course at Harvard that I took in 1964. But he never, ever, used even a page of it and I have never learned anything out of it, in spite of opening it several times over the years. His course resembled more the famous book by Dieudonne, Foundations of modern analysis, which I do like, although very abstract.

Years after I took Loomis's course I realized I had learned almost nothing there except for the very basic and important fact that a derivative should be thought of as a linear map, and the almost trivial fact that in infinite dimensional Banach spaces, not all linear maps are continuous.

What is the point of teaching differential calculus in Banach space if you never show anyone even one example of the derivative of an interesting function? E.g. since the space of paths has infinite dimension, a natural choice might be the derivative criterion for a smooth path between two points to have minimal length, confirming that a straight line is the shortest path between them. This example is a non trivial topic in its own right, the first one in the calculus of variations, treated in Courant, and nodded at in Loomis's book, (but not his course).

But if you are curious, look at all these books, including ones I did not like. We are all different, and you may appreciate something I did not. But I suggest you not pay a large sum of money for an out of print and rare copy sight unseen.

 

[mathwonk]

Spivak's Calc on manifolds is based on the idea tht the most important theorems are:
i) interchange of order of limits, ii) inverse function theorem, iii) fubini's theorem, and iv) stokes' theorem. so he does a careful job on all of those.

In fact maybe fubini is equivalent to change of order of limits, so there may be only three main theorems. I myself learned several variables calc from spivak's book.

 

[mathwonk]

here is a fair review of the nickerson et al. book by professional differential geometer:

http://projecteuclid.org/DPubS/Repo...w=body&id=pdf_1&handle=euclid.bams/1183523508and here is a rave review of another book i do not know, by barbara and john hubbard:

http://matrixeditions.com/VC.MAAreview.html

here is a copy of NCC for under $15:

https://www.amazon.com/dp/0486480909/?tag=pfamazon01-20
the comment I have always remembered by a famous mathematician about NCC was that it was concerned "more with the ride than the destination".

 

[jbunniii]

I haven't read it myself, but I've often seen this book highly recommended:

https://www.amazon.com/dp/1577663020/?tag=pfamazon01-20

Worth a look if you want a very classical treatment - no differential forms or even metric spaces.

 

[mathwonk]

a professor i respected very much used buck as his text for advanced calculus, and it is certainly one of the well regarded classical texts.

 

[sponsoredwalk]

Found another one:
Allendoerfer - Calculus of Several Variables & Differentiable Manifolds
Probably a bit too tough though but worth looking in

 

[wisvuze]

Found another one:
Allendoerfer - Calculus of Several Variables & Differentiable Manifolds
Probably a bit too tough though but worth looking in

I've looked through this one before. Nothing of particular interest: it's a fairly brief book with a horrible typesetting ( before LaTeX ). I would stick to Spivak's Calculus on Manifolds if you were going to resort to this book

 

[wisvuze]

When you reach the final chapters of Spivak's Calculus on Manifolds, a good book to read in conjunction is:
"Differential Forms" - Henri Cartan

 

[JG89]

I would recommend Analysis on Manifolds by Munkres. I have that book and Spivak's Calculus on Manifolds, but I find that Spivak's book is too terse.

 

[sponsoredwalk]

I've looked through this one before. Nothing of particular interest: it's a fairly brief book with a horrible typesetting ( before LaTeX ). I would stick to Spivak's Calculus on Manifolds if you were going to resort to this book

The second chapter is an exposition on existence & uniqueness theorems in ordinary
differential equations, proven both by Picard iteration & contraction maps, as well as
a discussion on global integrability criterion for vector fields & a generalization by Frobenius.
Third chapter contains two proofs of the inverse function theorem, one using Frobenius'
theorem. After that there's a hell of a lot of a discussion of manifolds I can't recall.
To say there's nothing of particular interest is a bit much, the typesetting is off-putting
but it's definitely worth a look if accessible.

 

[wisvuze]

The second chapter is an exposition on existence & uniqueness theorems in ordinary
differential equations, proven both by Picard iteration & contraction maps, as well as
a discussion on global integrability criterion for vector fields & a generalization by Frobenius.
Third chapter contains two proofs of the inverse function theorem, one using Frobenius'
theorem. After that there's a hell of a lot of a discussion of manifolds I can't recall.
To say there's nothing of particular interest is a bit much, the typesetting is off-putting
but it's definitely worth a look if accessible.

Hmm..I don't remember that stuff being in the copy I saw. Perhaps there was a difference of editions or my memory has just failed me, sorry. ( I don't mind bad type setting either, another good "advanced calculus" book is the one by Nickerson, except the typesetting is pre-LaTeX as well )

 

[sponsoredwalk]

No problem, dying to get the Nickerson book you mentioned.

 

[wisvuze]

No problem, dying to get the Nickerson book you mentioned.

If you didn't know, it has been recently released ( since September 2011 ) by Dover:

https://www.amazon.com/dp/0486480909/?tag=pfamazon01-20

So the book is now easily accessible!

( still waiting on Dover to hopefully release "Advanced Calculus" by Sternberg :( )

 

[battousai]

all things considered, would you guys say that munkres's book is better than spivak's?

 

[JG89]

^ I believe it's better. Spivak's book is very terse, while Munkres' goes into much more detail. Personally, I learned the subject from Munkres' book and will occasionally use Spivak's book as a reference regarding certain things. Also, I believe that Munkres' book covers more than Spivak's. For example, Munkres covers a little bit on abstract manifolds, deRham groups of punctured Euclidean space, etc. I don't think Spivak covers these.

I would advise one to go with Munkres

 

[wisvuze]

I think you would do well if you used both the books. If you want to "work for your education", spivak's book is more challenging ( if you do most of the exercises, also championing a terse page on material is part of the learning process ). The exercises in the Munkres book are either easier or computational in nature. I still think you should check out the Munkres book though, as it does contain many more details
(although, Munkres drops a couple hypotheses in some theorems to make the proofs easier )

 

[battousai]

yes, i have both books. browsing through the pages it seems like munkres devotes a chapter on reviewing linear algebra and other prerequisite topics, while spivak dives into the new stuff assuming the reader has a fair grasp and hence doesn't waste any time or pages for review.

i guess i can work on both at the same time? although i would guess that I'm on the side that needs to review the prerequisite topics.

 

[mathwonk]

If you are determined to get NSS (nickerson et al..) let me make a comment. They give there the wrong definition of the derivative. I.e. they focus on the directional derivative rather than the more important concept, the full derivative. NSS define on pages 146 and 172 the full derivative only in the case where the function has continuous directional derivatives. This does not even give the usual general definition of a derivative in the case of one variable.

The correct definition is that the derivative at x is a linear map L whose graph is tangent to the graph of the original function f, at x, after translation by f(x), in the sense that the limit

of [f(x+h)-f(x) - L(h)]/|h| is zero as h-->0.

This is made very clear in Loomis and Sternberg, Dieudonne, Fleming, and Spivak. The fact that the derivative can be deduced to exist and calculated from continuous directional derivatives, is a computational device that does not convey the real meaning of a derivative.

Another point is that the definition in NSS does not generalize to infinite dimensional spaces whereas the correct linear map version goes over word for word.

This is just one concrete example of why I feel NSS is not a good place to learn the subject. The facts are there, but the presentation does not make them clear to the student. This sort of unmotivated abstract teaching was very common in the 1960's.

The good part was the introduction of advanced topics and points of view. The weak point was the lack of motivation and examples. This was remedied to some extent in later works inspired by those early ones.

 

[sponsoredwalk]

If you are determined to get NSS (nickerson et al..) let me make a comment. They give there the wrong definition of the derivative. I.e. they focus on the directional derivative rather than the more important concept, the full derivative. NSS define on pages 146 and 172 the full derivative only in the case where the function has continuous directional derivatives. This does not even give the usual general definition of a derivative in the case of one variable.

The correct definition is that the derivative at x is a linear map L whose graph is tangent to the graph of the original function f, at x, after translation by f(x), in the sense that the limit

of [f(x+h)-f(x) - L(h)]/|h| is zero as h-->0.

This is made very clear in Loomis and Sternberg, Dieudonne, Fleming, and Spivak. The fact that the derivative can be deduced to exist and calculated from continuous directional derivatives, is a computational device that does not convey the real meaning of a derivative.

Another point is that the definition in NSS does not generalize to infinite dimensional spaces whereas the correct linear map version goes over word for word.

This is just one concrete example of why I feel NSS is not a good place to learn the subject.

After a long time I finally got my hands on a copy of the book today thanks to someone very helpful so I just have to ask you more about this, the back cover says:

"Starting with an abstract treatment of vector spaces & linear transforms, the authors introduce a single basic derivative in an invariant form. All other derivatives - gradient, divergence, curl & exterior - are obtained from it by specialization"

Now I see what you mean about them defining the directional derivative but how big of an issue is this as regards the rest of the content of the book?