What is considered Calculus on Manifolds?

[saminator910]

One can do calculus on a differentiable manifold, what does that mean? Does it mean you can use differential forms on the manifold, or that you can find tangent vectors, What is certified as "calculus on a manifold".

 

[Tenshou]

do you know about the calculus of index or index calculus?

 

[saminator910]

I am not familiar with index calculus.

 

[Tenshou]

It is just notation, I am no expert on neither manifolds or index calculus, but you are interested in them I suggest learning topology! Topology deals with manifolds, like $\delta$ complexes and betti numbers (which I recently started to read up on), also I would assume doing calculus on a manifold would be like doing arithmetic geometry(or something analogous to) it is just a way of understanding the "space" a little bit better. (I guess a way Einstein might put it). Sorry for not answering your question I am sure someone will come on here with a lot better answer than mine, till then cheers mate :D

 

[micromass]

do you know about the calculus of index or index calculus?

It is just notation, I am no expert on neither manifolds or index calculus, but you are interested in them I suggest learning topology! Topology deals with manifolds, like $\delta$ complexes and betti numbers (which I recently started to read up on), also I would assume doing calculus on a manifold would be like doing arithmetic geometry(or something analogous to) it is just a way of understanding the "space" a little bit better. (I guess a way Einstein might put it). Sorry for not answering your question I am sure someone will come on here with a lot better answer than mine, till then cheers mate :D

Honest question: do you think that these posts were helpful for the OP or do you just want to throw big words around?

 

[Tenshou]

I think it might help get a person interested in diving deeper into other things o.o' (I just got done reading/watching a video on this stuff by mathview on youtube), so yeah I think or thought that it could help o.o'

 

[dextercioby]

One can do calculus on a differentiable manifold, what does that mean? Does it mean you can use differential forms on the manifold, or that you can find tangent vectors, What is certified as "calculus on a manifold".

Calculus on a (differential) manifold is the extension of ordinary calculus in Rⁿ to more general structures, called differentiable manifolds. Rⁿ in an example of differentiable manifold, a simple one.

 

[saminator910]

I am familiar with topology, and differentiable manifolds. I realize technically while doing normal calculus it is on the manifold of Rn, its just fairly uninteresting. But how does one extend calculus to manifolds, I am very familiar with differential forms, but in the problems I've dealt with using differential forms, the manifolds have been parametrized and embedded in Rn, and by extension all manifold charts are practically parametrization. The fact is, this doesn't seem to be doing calculus on manifolds because they are embedded in Rn it is a very extrinsic view of the manifold, and really this is just normal calculus. Can you do calculus on manifolds that have been defined by patchwork instead of by charts?

 

[micromass]

Yes, it is entirely possible to do calculus on manifolds which are not embedded in \mathbb{R}^n. The idea is basically that you can do calculus in each individual patch, since it is it is diffeomorphic to \mathbb{R}^n. Then you just glue the results together. This is of course very informal, but that really is the basic idea. Lee's smooth manifold book deals with these kind of things very well.

 

[lavinia]

i learned these ideas in steps.

First one realizes that one can do calculus on an m-dimensional surface embedded in R^n. This is done by parameterizing the surface and taking directional derivatives along it. In fact all caclulus on manifolds can in theory be done this way. Calculus on the manifold is really just calculus on the parameter domain and is no different than calculus in Euclidean space.

This idea then generalizes and allows you to shed the ambient Euclidean space since all you really need is derivatives in the parameter domains. But you need to require that these operations do not depend on the particular parameter domain so it means that their overlap has to to be smooth. From this equality of derivatives in overlapping parameter domains - really just the Chain Rule - one gets the idea of the tangent bundle. The tangent bundle of a manifold in Euclidean space is just union of the tangent planes to the parameterized surface considered as a topological space.

One of the confusing things about doing calculus in Euclidean space is that it always uses the Euclidean inner product to translate differentials of function into inner products with vectors( the gradients of the functions). It is absolutely necesarry to understand that these inner products are not needed to do calculus. The abstract definition of manifold shows this.

 

[saminator910]

Can anyone suggest a good book on the topic? I am well versed in traditional vector calculus, as well as differential forms, and I have a good knowledge of linear algebra. I have also read a couple of books on point set topology, so I believe I have a solid grasp of that subject. Preferably I would like a book that is not all theory, and has some working examples. Would one go to a differential topology book for this subject? a differential geometry book? or a book specifically for the subject, such as Munkres Analysis on Manifolds, or Spivak's Calculus on manifolds? Be aware, I don't feel like I am yet operating on a graduate school level, so book choices may be limited.

 

[lavinia]

Spivak's book is a classic.

I assume with you full knowledge of vector calculus that you know how to do calculus on parameterized surfaces already. why not try a book on Riemann surfaces?

 

[mathwonk]

a manifold, such as sphere, differs from R^n in that it can have more than one choice of local coordinates near each point in which to check differentiability. thus the explicit numerical value of a derivative is no longer meaningful, but rather the question of whether or not a function is differentiable is still meaningful. it i also not meaningful to integrate a real valued function on a manifold, but differential forms are a variation on real valued functions, being functions whose values are "covectors" on the manifold, and these can still be integrated over curves.

 

[saminator910]

Do you mean Riemann surfaces, the complex manifolds, or Riemann manifolds, manifolds with Riemann metrics, if you do mean Riemann surfaces, how accessible is the topic to someone with a minimal background in complex analysis?

 

[mathwonk]

only you can answer the question of how accessible something is, by trying to access it yourself.

 

[lavinia]

Do you mean Riemann surfaces, the complex manifolds, or Riemann manifolds, manifolds with Riemann metrics, if you do mean Riemann surfaces, how accessible is the topic to someone with a minimal background in complex analysis?

You need to know some complex analysis for Riemann surfaces but if you already know vector calculus you should learn it. Calculus on surfaces is extremely accessible and gives intuition for other cases. If you do not want to learn complex analysis - a mistake in my opinion- then I always recommend the book by Singer and Thorpe called Introduction to Elementary Topology and Geometry. It focusses on surfaces and derives differential geometry completely intrinsically through connections on the tangent circle bundle. It also teaches general calculus on manifolds including de Rham's theorem. I can't recommend this book enough. It is written for undergraduates who want to learn modern geometry and algebraic topology.

 

[Jim Kata]

What it means is you have a pullback to R^N. Calculus on a manifold means you can pullback to a subspace of R^n and do your integral there

 

[D H]

Calculus on a manifold means you can pullback to a subspace of R^n and do your integral there

You can do your derivative there. Integration is a rather different beast.

There's been a lot of work in the last 15 or so years on how best to do numerical integration on a manifold, Lie groups in particular.

Naively use the numerical integration that work so nicely on R^n on on a Lie group as if it were R^n and you'll quickly run into trouble. The very first step takes you off the manifold, and subsequent steps make things that much worse. For a long time, the standard kludge was to pull each step back on to the manifold.

This is a kludge, and it is essentially invalid. That it happens to work (kinda, sorta) is a feature of the fact that locally a manifold looks like R^n.

Much better is to use some technique that keeps the integration on the manifold. There has been a lot of work on such techniques as of late, at least for Lie groups. All of them use the exponential map. Some also use the commutator, some use this nasty thing called d\exp^{-1}, or dexpinv.