Manifolds, coordinate systems, books

Question

Which books, say Lee's Introduction to Smooth Manifolds or Munkres' Analysis on Manifolds explains how the theory of a differentiable manifolds can be used to solve a problem that is expressed in a spherical coordinate system? Such a problem could be calculation of volume of a sphere, or Maxwell speed distribution summing over all directions, while using manifolds. Do any of the books listed provide such an explanation?

Answer 1

Spherical coordinates are native to $R^3$ and cannot directly be generalized to higher dimensional spaces. If you look at how this coordinate system is defined, you'll see that it relies intimately on the local geometry of three dimensional Euclidean space.

Now there are certainly higher dimensional manifolds that are locally homeomorphic in their three-dimensional subspaces to $R^3$ under spherical coordinates, but this then requires the construction of not only an appropriate family of functions to define the coordinate system, but the local subspace itself where this coordinate system is valid. Tensors are usually needed to make this constructions precise. This is why many differential geometers outside of physics work in a coordinate-free manner via differential forms,looking at the "big picture" on the manifold rather then specific coordinate systems. Of course, this doesn't make coordinate systems any less important on manifolds -- particularly in modern mechanics.

---

Now comes the sales-pitch:

I would recommend $1$ specific book in particular for you to begin to study coordinate systems on abstract manifolds and it's relatively cheap. The book is Differential Topology: An Introduction by David Gauld. This is a strange, almost unknown textbook Dover unearthed -- it was first published 30 years ago and went out of print almost as fast.

It's an introduction to smooth manifolds aimed at sophisticated undergraduates. It’s prerequisites are small: just a rigorous course in one variable analysis using $\epsilon$-$\delta$ arguments.

The book has a lot of good pictures, mostly careful rigor and explains many things very well-it also has a lot of nice exercises, none too hard. The book’s real strength -- something no other introductory book on differential manifolds do, to my knowledge -- is the hundreds of specific examples of local coordinate systems as differentiable structures via charts and the very precise formulation of orientability on these structures in Euclidean space. No book does this comprehensive a job of giving examples of these 2 critical concepts and for that reason alone, the book is worth having.

The main problem with the book is that Gauld tends to lapse into idiosyncrasies of language that make the book very confusing at times -- and it’s jarring given what a good job he does with the rest of the book. The main example of this is in the opening 3 chapters which develop the necessary point set topology needed for the book -- and he does it in terms of the old 1970’s paradigm of “nearness” i.e. topological properties are developed entirely from axioms derived from the properties of limit points. For students with either minimal backgrounds in topology or those that have been trained in the “usual” formulation of topology, these chapters may be utterly baffling. Fortunately -- this material plays no role in most of the rest of the book, it’s merely so that differentiable manifolds can be defined precisely.

He also does a good job explaining surgery on low dimensional manifolds, a better job then most books do. Unfortunately, he also does a sloppy job of using it to prove the classification theorem of surfaces, with a lot of handwaving.

That being said -- most of his presentation is excellent and very useful and at this price, you really can’t complain too much.That book should help answer a lot of your questions.