Calculus Practical reference for integration on manifolds

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The discussion centers on the need for practical examples of integrals over surfaces and volumes in general relativity, particularly for understanding concepts like pulling back metrics onto submanifolds and working with normal vectors. Eric Poisson's "A Relativist's Toolkit" is recommended for its practical insights, especially Chapter 3 on hypersurfaces. While some participants mention other resources, such as books by Frankel, Fecko, and Darling, there is a consensus that finding comprehensive examples is challenging. Orodruin's book is also suggested for its chapter on calculus on manifolds, which may provide additional insights. Overall, the focus is on bridging theoretical knowledge with practical application in the context of general relativity.
etotheipi
I was trying to look for something that works a lot of examples of integrals over surfaces, volumes etc. in general relativity. Tong's notes and some others are good on the abstract/theoretical side but it'd really be better at this stage to get some practice with concrete examples in order to see how everything fits together. Does anyone know a good place? Thanks
 
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etotheipi said:
I was trying to look for something that works a lot of examples of integrals over surfaces, volumes etc. in general relativity.
This probably is not what you have in mind, but have you ever looked at "A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics by Eric Poisson?

Videos by Poisson (look at google drive link):
 
Thanks, I hadn't come across that before but it looks like a nice set of videos.

What I'm looking for specifically is to get a feel for how in practice you go about doing things like pulling back a metric onto a submanifold, working out what the normal vectors are, and just generally converting integrals on general manifolds into doable integrals on ##\mathbf{R}^n##.

I think I have a vague, basic (very non-rigorous) idea of the theory, but struggle with the subtleties and figured some concrete examples might help.
 
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etotheipi said:
What I'm looking for specifically is to get a feel for how in practice you go about doing things like pulling back a metric onto a submanifold, working out what the normal vectors are, and just generally converting integrals on general manifolds into doable integrals on ##\mathbf{R}^n##.

Poisson does this in chapter 3 "Hypersurfaces" of his book, without mentioning, e.g., pullbacks.

I can't think of books that have loads of examples. I know of differentiable geometry books that present the theory in a readable manner (e.g., books by Frankel and by Fecko), but I am not sure how many examples that they present. Another possibility (which I haven't look at in 20 years) is a book on differential forms by David Darling.
 
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George Jones said:
Poisson does this in chapter 3 "Hypersurfaces" of his book, without mentioning, e.g., pullbacks.

I can't think of books that have loads of examples. I know of differentiable geometry books that present the theory in a readable manner (e.g., books by Frankel and by Fecko), but I am not sure how many examples that they present. Another possibility (which I haven't look at in 20 years) is a book on differential forms by David Darling.

Thanks, that sounds great, I'll take a look at Poisson's book!
 
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caz said:
You might check out @Orodruin ’s book
https://www.amazon.com/dp/1138056901/?tag=pfamazon01-20
It has a chapter on calculus on manifolds. Even if it is not what you are looking for, he might have some ideas on other places to look.

Question: when the table hits 12:00 are you finished with Wald?

I do actually have @Orodruin's very nice book, it's my go-to maths methods reference! There are some relevant problems at the end of that chapter which I haven't tried yet.

(Also, I'm a bit sleepy at the moment so I don't quite understand the last sentence, haha... but I think it's fair to say I will never be finished with Wald 😵)
 
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etotheipi said:
(Also, I'm a bit sleepy at the moment so I don't quite understand the last sentence, haha... but I think it's fair to say I will never be finished with Wald 😵)

The table is rotating. I was trying to impose order upon it. Vertical table implies that you have become one with Wald.
 
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