Book on differential geometry

  • Thread starter Telemachus
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  • #26
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The book by Singer "Symmetry in Mechanics" tries to introduce symmetries in the manifold setting but in a way that doesn't assume much math (she doesn't properly define topological manifolds and is pretty hand wavy on lie groups and lie algebras and group actions and their orbits even though she then uses these concepts to talk about conserved systems) however, as noted, the lack of mathematical rigor will leave you unsatisfied and/or confused in my experience.
Ok! That seems to be slightly more up my alley, and the group theory would definitely help if I end up taking QFT in grad school (still waiting on decisions, fingers crossed). Something like that but without the hand-waving?

I can deal with rigor, I just don't want something that requires proving every important non-trivial theorem to finish it. Like I said it's for self-learning and fun, not for reference/a rigorous course.
 
  • #27
WannabeNewton
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Ok! That seems to be slightly more up my alley, and the group theory would definitely help if I end up taking QFT in grad school (still waiting on decisions, fingers crossed). Something like that but without the hand-waving?

I can deal with rigor, I just don't want something that requires proving every important non-trivial theorem to finish it. Like I said it's for self-learning and fun, not for reference/a rigorous course.
I mean if you like it then use it, everyone has their own tastes and preferred mathematical expositions when it comes to physics texts. In my experience, I am absolutely terrible at interpreting math as presented in physics texts and usually need to go to a math textbook on the subject in order to free up confusions. You probably get the math as presented in physics texts very easily so all the power to you. See if you can check out Singer somehow and judge for yourself if you like it or not but it definitely covers some of the things you are interested in regarding dynamics of N particle systems. I don't know any QFT or QM at all so I don't know if the level of group theory presented in Singer will help you for QFT or QM unfortunately, sorry about that :[. Good luck with the grad school business! I'm sure you'll get in love =D.
 
  • #29
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I'm really surprised no one has mentioned Gauge Fields, Knots and Gravity by John Baez...
https://www.amazon.com/dp/9810220340/?tag=pfamazon01-20&tag=pfamazon01-20
I love that book, it's not very rigorous, but it provides a lot of insight, and for each topic gives a list of references, so that if you wish to, you can check the proofs and stuff in pure math books.

A slightly more rigorous book (and also a very good one), is The Geometry of Physics by Frankel
https://www.amazon.com/dp/1107602602/?tag=pfamazon01-20&tag=pfamazon01-20
It gives lots of applications to GR, CM, continuum mechanics, etc.
 
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  • #30
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A slightly more rigorous book (and also a very good one), is The Geometry of Physics by Frankel
https://www.amazon.com/dp/1107602602/?tag=pfamazon01-20&tag=pfamazon01-20
It gives lots of applications to GR, CM, continuum mechanics, etc.
Wow I think you are spot on with this one, looks like it has everything I wanted and more, the preview looks good. It's a bit more expensive than what I was hoping for, but from the looks of it I am probably going to get much out of it. Does it have solutions for some of the exercises?
 
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  • #31
WannabeNewton
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Does it have solutions for some of the exercises?
Unfortunately it doesn't the last time I saw it which was in a course on differential topology for physicists. The section on the connections between circuit theory and topology was pretty mind blowing though.
 
  • #32
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Does it have solutions for some of the exercises?
I have the second edition and it doesn't contain solutions. Maybe they added some in the third edition but I'm not sure...
 
  • #33
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Did this hamper you at all using it as a sole resource for learning the contained subjects? Ie: are the problems more or less straightforward applications of the preceding theory or easy to infer their correctness, or are you left in the dark to toil like with some other books (Goldstein, Landau, etc.)?
 
  • #34
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In my opinion the exercises are not that hard. I think there's a good balance between computational exercises and proofs... you souldn't have much trouble figuring out whether your answers are correct or not.
 
  • #35
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This is a joke right? Are you actually trying to help the OP learn or just throw really hard books for no reason at someone looking for an introduction?
Sometimes I think the people replying are just trying to show off and "overkill" instead of actually helping the OP.
 
  • #36
lurflurf
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Like in the other thread I recommend to toe dippers with "some" calculus and linear algebra and an interest in physical stuffs Curvature in Mathematics and Physics by Shlomo Sternberg. To the sentimentalist who think three dimensions is more than enough try differential Geometry Of Three Dimensions by C.E. Weatherburn.
 
  • #37
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Sometimes I think the people replying are just trying to show off and "overkill" instead of actually helping the OP.
In the guy's defense, I didn't feel like I understood differential geometry at all until I borrowed a copy of volume 1 of Spivak's introduction. It opens by defining a manifold as a metric space rather than the more general topological space, but other than that it was fantastic. Plus, I like the way Spivak writes. I don't know about anyone else, but it often reads like he's talking to you rather than attempting to talk around your perceived skill level. I appreciate this.

That being said, Spivak doesn't hold much back either. :tongue:
 
  • #38
WannabeNewton
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Spivak isn't a problem. Lots of people use Spivak for a first exposition to differential topology. Only Gauss, Riemann, or Weyl would use Lang as an intro to the subject.
 
  • #39
lurflurf
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^I don't see any problem looking at Lang (or Kobayashi and Nomizu) early on. It does not cause your face to melt. Lang is one of the few books with infinite dimensional flavor, and as is often the case with Lang, he presents things as they are best understood instead of easiest understood. Still most people would like to also read a more gentle book. Spivak is pretty chatty which others dislike, but I consider it a strength. I dislike the typeset though and if I recall correctly it is unchanged in the third edition.
 
  • #40
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The issue I have with Lang is it has no exercises. Otherwise I think it would be a reasonable choice for a dedicated student.
 
  • #41
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^I don't see any problem looking at Lang (or Kobayashi and Nomizu) early on. It does not cause your face to melt. Lang is one of the few books with infinite dimensional flavor, and as is often the case with Lang, he presents things as they are best understood instead of easiest understood. Still most people would like to also read a more gentle book. Spivak is pretty chatty which others dislike, but I consider it a strength. I dislike the typeset though and if I recall correctly it is unchanged in the third edition.
The issue I have with Lang is it has no exercises. Otherwise I think it would be a reasonable choice for a dedicated student.
Man, even my old differential geometry professor said that he looked at Lang and didn't understand much of it because it was so horrible written. And this is a guy who knows differential geometry inside out.
 

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