What is "Analysis on Manifolds"?

[David Carroll]

Greetings. I just bought a textbook and I have no idea what it is about. A little explanation is in order:

One of my goals in life has been to obtain a degree in mathematics. Unfortunately, I have made very poor life choices that have made this goal practically unachievable, which I won't detail but if I say one of those choices was to spend student loans to support a heroin addiction, you get the basic idea...but what has been has been. I cannot change the past nor its consequences.

Nevertheless, I have made the decision to obtain the epistemological equivalent of a degree in mathematics by perusing the course listings for the math department of the University of Michigan, buying the associated textbooks, and learning the subjects by myself. I have made it through Calculus I, II, and III and I am in the middle of Linear Algebra. I have a textbook for "Boundary Value Problems and Differential Equations" that awaits me after I am done with Linear Algebra.

I noticed that the next suggested course after differential equations was called by the U of M "Analysis". I went ahead and bought its first associated textbook "Analysis on Manifolds" by Munkres and it is in a facility somewhere, waiting to reach me via U.S. Postal Service and by courtesy of amazon.com

The difference between this course and the others was that if someone were to ask me, before I ever learned any of this stuff, "What is Calculus? What is Linear Algebra? What are differential equations?", I could at least give a rough, vague outline of what they were. I just could not perform any calculations of what I just described...until now.

Not so with this course known as "Analysis".

So out with the obvious question, "What on Earth IS analysis and what IS a manifold?"

 

[Fredrik]

Analysis is essentially calculus, with emphasis on proofs instead of on how to calculate stuff. A manifold is essentially just a set M together with a bunch of functions $x:U\to\mathbb R^n$ with $U\subseteq M$, all with the same n. These functions are called coordinate systems or charts. They can be used to define partial derivatives of functions $f:M\to\mathbb R$ even when there's no addition operation on M.

 

[micromass]

If you have never done analysis before, then it is probably not wise to go through Munkres. Analysis is basically calculus made rigorous (it goes beyond calculus, but let's take this description for now). Munkres is basically calculus 3 made rigorous. So it would be wise to go through a good single variable calculus text first before tackling Munkres.

Anyway, what are manifolds? Up to now, you have done much calculus in $\mathbb{R}^n$. Specifically, you know how to do calculus on $\mathbb{R}^2$. You know:

• how to find tangent lines to curves
• how to calculate shortest distances between curves in $\mathbb{R}^2$
• how to solve trig questions on $\mathbb{R}^2$
• how to calculate surface areas, volumes, and so much more.

This is good. But when applying this to our own lifes, it is useless, since we don't live on in a flat plane, we live on a sphere. And do you know how to do all the above things on the sphere? I guess you don't (or not without giving it quite some thoughts). Furthermore, general relativity states that our universe isn't even flat, it is curved. So somehow, we want to do calculus (i.e. doing the above four things) on more general spaces than flat spaces. So is entire calculus on $\mathbb{R}^n$ useless? No. Because you can make the observation that when we live on our earth, we can treat it as flat. Indeed, when calculating the area of a piece of land on earth, we just assume that the Earth is flat and get it over with. So locally, we see no difference between our Earth and $\mathbb{R}^2$. So this gives a hint on how to apply calculus tools on our earth. And this is where manifolds come in. Manifolds are a vast and powerful generalization of these techniques. A manifold is something such that somebody who lives on it sees no difference between the (perhaps curved) manifold and $\mathbb{R}^n$ for some $n$. This allows us to do calculus on it.

 

[verty]

I cannot change the past nor its consequences.

This is a bad point of view to have. Have you heard of the sampling bias? It's the argument that if one surveys a population by taking a sample, and one tries as much as possible to select people in a random way, it will still be the case that something ties the sampled people together. It may be that they live in one city, that they are they type of people who answer surveys, or that they tend to be middle-aged.

Because there will always be (it is supposed) something tying together the people in the sample, there is a bias versus the population itself. The population can not be assumed to resemble the sample in every way. That is, you can never assume that what is true for the sample is true for the population in toto. There will always be, according to this idea, some way in which the sample differs.

And why wouldn't the recent past be a sample based on a contingent series of circumstances that at any time could change? You can't change the past, that is of course true, but you don't know what the future holds, you can't know because this idea applies. The past is a sample of your lifeline and the the future will differ from it, that I guarantee.

Nevertheless, I have made the decision to obtain the epistemological equivalent of a degree in mathematics by...

Why did you use this big word? You were speaking to a general audience, asking a question that you wanted many people to read, and you used this odd language that no one understands. You also bolded "equivalent". It's even debatable whether it is equivalent. And anyway, an epistemological equivalent would be something like a bible commentary which was equivalent to reading the passage itself. Are you assuming that you will of course be able to gain just as much knowledge yourself as you would in a class? And by your bold text, I read it as given that you will be able to do that. It just seems odd because you at once assert both that you can't change the future, you have no power to do that, but also that you are some kind of gifted soul who doesn't need no college.

So I just mean to point out that you need to come down to Earth a little bit there, bud. Maybe the future can change, maybe you don't know it all, you know?

PS. I apologize if this was a bit blunt. Perhaps I could have said it more delicately, I don't know. I will PM Mr Carroll with some additional comments but I won't do that here.

 

[David Carroll]

I'm not that gifted, no. I just assumed that the purpose of a teacher/professor was to help students master the subject matter which already exists in the textbook (unless, of course, it's a class with no textbook). And if I can do that without the benefit of the teacher, more power to me, I suppose. Although one would rightfully object that I had no qualifications to become a math teacher even if I obtained this "epistemological" equivalent. But becoming a teacher is not one of my goals. Simply mastering the subject matter is.

If your implication was that I come off as pompous in my posts, I don't mean it to be condescending. It is an opinion held by many that people should write the way they talk. And if I were to meet you, this is exactly how I would talk. So I'm not insincere, just autistic...-ish.

 

[Fredrik]

Some people are able to save a lot of time by learning important things from the lectures. Other people find it difficult to learn anything from lectures. University studies are easier for the former, but it's not like everyone who's studying at the university belongs to that group. The ones who don't, have to do essentially the same thing as you. I see no reason to think that you can't do it. I'm more concerned about the fact that you will spend an enormous amount of time on this, and then you probably won't be able to use it for anything. You may find it satisfying in some ways, but it won't get you into grad school and it won't get you a good job.

 

[verty]

I'm not that gifted, no. I just assumed that the purpose of a teacher/professor was to help students master the subject matter which already exists in the textbook (unless, of course, it's a class with no textbook). And if I can do that without the benefit of the teacher, more power to me, I suppose. Although one would rightfully object that I had no qualifications to become a math teacher even if I obtained this "epistemological" equivalent. But becoming a teacher is not one of my goals. Simply mastering the subject matter is.

How do you assess your chances of mastering the material in Analysis on Manifolds? That is, what would the odds need to be for it to be a good bet that you would master that material? My guess would be, something like 8-1. At 8-1, it seems like a good bet that you would succeed. At 7-1, it seems like a bad bet, at least to me. How do you rate it?

If your implication was that I come off as pompous in my posts, I don't mean it to be condescending. It is an opinion held by many that people should write the way they talk. And if I were to meet you, this is exactly how I would talk. So I'm not insincere, just autistic...-ish.

 

[David Carroll]

Well, I think similar chances held for my ability to learn calculi 1 to 3 (there's some pompous language: pluralizing calculus as "calculi" ) and linear algebra. Since I've succeeded so far, I think my chances of mastering analysis should be more optimistic than otherwise.

 

[verty]

Well, I think similar chances held for my ability to learn calculi 1 to 3 (there's some pompous language: pluralizing calculus as "calculi" ) and linear algebra. Since I've succeeded so far, I think my chances of mastering analysis should be more optimistic than otherwise.

I should explain that my odds of 8-1 were based on the following situation: you read Analysis on Manifolds as soon as it arrives, you get through it in reasonable time with minimal use of other sources (including any single-variable book), and by the end you can with say with reasonable certainty that you have mastered the contents. I think I'd even take your bet at 7-1 that you couldn't do it. But if you can then props to you.