- #26

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Edwards' book looks quite appealing too, I wish I could get a look at both books in person.

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- #26

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Edwards' book looks quite appealing too, I wish I could get a look at both books in person.

- #27

Landau

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No, it doesn't. You should be able to tackle Munkres perfectly!Thanks for the tips. I've been through a first-semester real analysis course, a linear algebra course, and I've had some basic topology and metric space theory. Does Munkres' demand more prereqs than this?

- #28

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That should be enough for Munkres' book. You can browse a limited preview of either book on Google Books. This is a great resource. Another thing is that both of these books are available (although illegally) online through either .djvu or .pdf files. I use this to help decide which book I need on a specific topic, if a book has the topics I want, and if I even like the book. Then I usually check out the book from the library or buy the book to support the author. Plus it's difficult to read a math text on the computer and expensive to print it out.

Edwards' book looks quite appealing too, I wish I could get a look at both books in person.

- #29

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An old-fashioned index nightmare approach. Not that there might not be insights to be found here, but it's pretty hard on the eyes. Older books that are still readable are Flanders and Bishop & Goldberg.I am curious if anyone has looked at:

Tensors, Differential Forms, and Variational Principles

by Lovelock and Rund

https://www.amazon.com/dp/0486658406/?tag=pfamazon01-20&tag=pfamazon01-20

Not mentioned so far is Darling's

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- #30

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

A question to the orthers: a lectures notes circulate in the web that were then turned into David Bachman's book. How complete are they? What's more in the book?

Thanks.

Goldbeetle

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- #32

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http://www.math.cornell.edu/~sjamaar/classes/3210/notes.html

is very easy. Of course, if it's that easy, then maybe it's too watered down?

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RedX Thanks,

iamthegelo, I'm also interested in a solution manual for that book if it exists

iamthegelo, I'm also interested in a solution manual for that book if it exists

- #34

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Thanks for the feedback. I will start with the Bachman pdf instead and then see if I should get the hard copy.An old-fashioned index nightmare approach. Not that there might not be insights to be found here, but it's pretty hard on the eyes.

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I feel compelled to resurrect this great thread.

https://www.amazon.com/dp/0127425101/?tag=pfamazon01-20&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.

I am assuming you mean this:...by all means have a look also at the excellent "Differential Forms" by Steven Weintraub.

https://www.amazon.com/dp/0127425101/?tag=pfamazon01-20&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.

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- #38

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Hey, I'm currently working through https://www.amazon.com/dp/0716749920/?tag=pfamazon01-20&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&tag=pfamazon01-20

(the contents are here), it looks like it'd be a good primer for Hubbard's book, what do you think?

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&tag=pfamazon01-20

(the contents are here), it looks like it'd be a good primer for Hubbard's book, what do you think?

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- #41

mathwonk

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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.)

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.)

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- #42

Tom Mattson

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Yeah I remember that. I actually got the impression that you didn't like the book. Glad to hear otherwise!One of my faults is focusing on the few negative aspects of a situation instead of the many positive ones.

- #43

mathwonk

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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.

- #44

mathwonk

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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.)

- #45

mathwonk

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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.

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

- #47

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

- #48

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You should try Spivak's Calculus on Manifolds.

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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.

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