Practical reference for integration on manifolds

[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

 

[George Jones]

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):

 

[etotheipi]

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.

 

[George Jones]

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.

 

[etotheipi]

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!

 

[Frabjous]

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?

 

[etotheipi]

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 )

 

[Frabjous]

(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.