Which Books Are Best for Physicists Learning Manifolds and Differential Forms?

[Sankaku]

I feel compelled to resurrect this great thread.

...by all means have a look also at the excellent "Differential Forms" by Steven Weintraub.

I am assuming you mean this:
https://www.amazon.com/dp/0127425101/?tag=pfamazon01-20

I just found an ebook version of this fabulous text.

I really like his approach - it is strangely surprising when you find an author that makes complete sense. I am sure that other books mentioned in this thread will be more rigorous and go deeper that Weintraub, but it is a great introduction for those who have multi-variable Calculus under their belt.

 

[Goldbeetle]

Yes, indeed, that is the book! You can complement/compare it with the notes you find at the link posted by RedX.

 

[sponsoredwalk]

Hey, I'm currently working through https://www.amazon.com/dp/0716749920/?tag=pfamazon01-20
and once I finish these I'd like to read a book on differential forms tying it all together.
I've read this thread and I like the suggestions but I found a book that takes a slightly
different approach, I'd just like some input from you guys on it: https://www.amazon.com/dp/047152638X/?tag=pfamazon01-20
(the contents are here), it looks like it'd be a good primer for Hubbard's book, what do you think?

 

[Daverz]

The books that annoy me are the ones that start out by cheerfully defining forms as just the things that go under the integral. Head smack desk.

 

[deluks917]

I don't know anything about Shifrin's book but I'd avoid Hubbard's. Hubbards book was written for a freshmen class that had only calc BC and had probally never seen much proofs before. If you fit this description get the book. However the book is slow going. Its "main part" is 660 pages and yet the most important proofs are in the appendix (change of variable,implicit function, stokes). Worse the proofs in the text and in the appendix are amzingly long, Hubbard actually takes 12 full pages to prove Stokes theorem. I would stronbgly recommend reading either Analysis on Manifolds by Munkres or Calculus on Manifolds by Spivak. Munkres actaully treats the subject in greater depth in 300 pages then Hubbard does in 812. I understand why he wrote the book the way he did but I can't recommend it.

 

[mathwonk]

N!kofeyn, my apologies for "running off" David Bachman. I think that is inaccurate, but i do recall he asked for corrections on his book and when I took him at his word and pointed out some mathematical errors I had noted, he got angry. I think he stuck around longer than I did however, so you might say he ran me off. I could be wrong.

Anyway I think his book is excellent, the best place I know of to get a real feel for the geometric meaning of differential forms (as measures of oriented volume of "blocks"), and I wish I had refrained from pointing out what were in reality very tiny and subtle errors that would not hinder any student from benefiting from his book. Such errors exist in almost all books, even some of the best and most useful. Lang's famous algebra book abounds with them, as do many other famous and helpful books.

Lesson learned: when people ask for criticism, they usually do not mean it, they really want praise. (Me too.) One of my faults is focusing on the few negative aspects of a situation instead of the many positive ones. This sort of perfectionism makes it hard to actually produce any creative work, and should be avoided as much as possible. Or at least avoided until the last step. After producing a creative work it seems useful to me to go back over it and correct the errors. But when pointing these out in the works of others it helps to be very diplomatic. Producing a creative work takes a lot of effort and displaying it to others afterwards also takes courage, and we should be grateful to these people for helping us learn from them.

Many people including myself, write books which even if they are correct and free from serious false statements, still may have limited usefulness because they are not illuminated by any deep understanding of the subject we are writing about. It is probably thus better to read books by people who really know something worth learning from them even if their treatment contains errors. My notes on the Riemann Roch theorem for example were written before I understood the topic. Still in trying to write up the subject I eventually came to feel I understood it. The main breakthrough was reading Riemann when a translation into English became available.

An outstanding theoretical book on differential forms at least for math students (like myself) is the one by Henri Cartan, all of whose writings are to me a model of perfection, i.e. clear, correct, and succinct. This one is also available in a cheap paperback.

Oh, and Loring Tu is especially famous as a writer whose works are models of clarity. It helps of course to know his clearly stated prerequisites. Still I would suggest one can always learn some thing from Loring.

If you want to share my attempt at explaining something I did not understand at the time fully, there is a section in my free (you get what you pay for) web page algebra notes (math 845-3) near the end, that treats "exterior algebras", i.e. the algebraic aspect of differential forms.

http://www.math.uga.edu/~roy/ (that young kid there was apparently me a longgg time ago.)

 

[quantumdude]

One of my faults is focusing on the few negative aspects of a situation instead of the many positive ones.

Yeah I remember that. I actually got the impression that you didn't like the book. Glad to hear otherwise!

 

[mathwonk]

daverz, those books are telling you the properties that a form should have. i.e. a one form should see a curve and spit out a number, and it should spit out the same number when you change the parametrization of the curve.

this suggests how to define them. I.e. given a smooth curve, you get a smooth family of tangent vectors. hence you get a number in two steps. First by assiging a number to each tangent vector you get a function, then by integration you get a single number.

so you want something that assigns a number to a tangent vector, hence a differential form should be a function on tangent vectors. second, you want the number to be the same when you reparametrize the curve, so you want the fuynction on tangent vectors to get larger when the tangent vector gets larger, i.e. when you run over the curve faster, hence =integrate over a shorter interval, you want to compensate by getting a larger function. hence you want a differential form to be a LINEAR function on tabngent vectors. hence a differential form is a family of linear functions on tangent vectors, i.e,. a "covector field".

or simple mindedly all you need to know about a one form df is that df/dz dz = df. I.e. they are the things that go under integral signs and justify the usual rules for change of variables in integration. since an integral changes by the jacobian determinant of the change of variables, so should the differential form change that way, i.e,. a "form" should be (multi)linear and alternating, like a determinant.

 

[mathwonk]

I first got over my fear of differential forms from an article by harley flanders in an MAA book on global differential geometry by ?Chern et al? where he just started calculating with them. I mean who cares if a differential form is a field of alternating n-linear covectors, if all you need to use them is to know that
dx^dy = -dy^dx, hence dx^dx = 0? (and scalars pull out too.) Hence

(dx+dy)^(dx+dy+dz) =

dx^dx + dx^dy + dx^dz + dy^dx + dy^dy + dy^dz =

0 + dx^dy + dx^dz -dx^dy + 0 + dy^dz =

dx^dz + dy^dz.

and more generally (a.dx+b.dy)^(c.dx+d.dy) = (ad-bc)(dx^dy).

so try (adx + bdy + cdz)^(d.dx+e.dy+f.dz)^(g.dx+h.dy+i.dz) = ?

(hint: it should look like a 3by3 determinant.)

 

[mathwonk]

I have not read bamberg and sternberg, but i know who they are. sternberg is a brilliant differential geometer at harvard for a long time, and bamberg is a physicist who was considered almost the only excellent teacher in the physics department in the 1960's. He also ahd a sense of humor. I still have a copy of his lab guide to a new piece of equipment, the "turbo encabulatior" fully equipped with something like quasiboscular grammeters tankered to the bendyles.

by scanning their list of supplementary reading this is apparently an advanced book, for harvard students, since they mention loomis and sternberg as parallel reading, and I am gratified to learn they recommend also several of my favorite books on de, such as arnol'd and braun. they also recommend hirsch and smale, which i thought was not well written.

this brings up an unfortunate fact of life. when outstanding researchers take time to write a book, they do not always want to expend adequate time to make it perfectly written. we have to accept flaws in exposition in exchange for the wonderful insights they offer that go beyond those possible for ordinary authors. that may be the case here, although i would have thought bamberg's reputation as great teacher would have mitigated such problems. this applies perhaps to hirsch and smale. I.e. they are tremendous authorities, but i found their book somewhat carelessly written. perhaps i was wrong. arnold's books are both insightful and well written however, as is braun's.

 

[n!kofeyn]

Hey mathwonk, I apologize for my comments in that thread. I was obviously a little angry at something and let my anger flow in an incorrect manner based upon that one thread. I'm sorry for the ad hominem attacks. I remembering being frustrated with some unhelpful/condescending threads in general, as well as a few threads that were shutdown that contained legitimate questions from someone who was curious. Some of my comments may have been valid, but they were marred by silly attacks, and I ended up taking a high brow approach myself, which was the very thing I was attempting to condemn.

Either way, it's great to see so much interest in differential forms!

 

[PhizKid]

I don't think he's still angry over what you said 2 years ago, much less remember it...

 

[kostas230]

You should try Spivak's Calculus on Manifolds.

 

[Mike706]

There is also A Geometric Approach to Differential Forms by David Bachman. I didn't know which heading to fit it under. :) There is actually a thread here where someone wanted to get a group to go through the book and in which Bachman took part in, until mathwonk ran him off.

I just wanted to give a +1 for this book. It's short, and doesn't take much time to work through. It gives a good intuitive understanding of forms, and I would read this book to get a feel for the subject before starting a more advanced and/or rigorous study.