# Practical reference for integration on manifolds

• Calculus
• etotheipi
In summary, the conversation revolved around finding resources for practicing integrals over surfaces and volumes in general relativity. Suggestions were made for Poisson's book, @Orodruin's book, and a book on differential forms by David Darling. The conversation also touched upon the use of pullbacks and working with manifolds in practice. The question of when one will be finished with Wald was also brought up.
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

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?

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.

vanhees71
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.

vanhees71 and etotheipi
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!

vanhees71, Demystifier and etotheipi
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 )

vanhees71
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.

vanhees71 and etotheipi

## 1. What is the purpose of integration on manifolds?

The purpose of integration on manifolds is to calculate the total value of a function over a specific region on a manifold. This can be used to solve various problems in fields such as physics, engineering, and mathematics.

## 2. What is a manifold?

A manifold is a mathematical concept that describes a space that locally resembles Euclidean space. In simpler terms, it is a curved surface that can be represented by a set of coordinates.

## 3. How is integration on manifolds different from integration on regular surfaces?

Integration on manifolds is different from integration on regular surfaces because the integration is done over a higher-dimensional space. This means that the integration must take into account the curvature of the manifold, which requires more advanced techniques and formulas.

## 4. What are some real-world applications of integration on manifolds?

Integration on manifolds has many real-world applications, including calculating the center of mass of an object, finding the volume of a curved surface, and determining the path of a moving object in space.

## 5. What are some common techniques used for integration on manifolds?

Some common techniques used for integration on manifolds include Stokes' theorem, Green's theorem, and the divergence theorem. These techniques allow for the integration of vector fields and differential forms over a manifold.

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