Calculus on Manifolds - operational point of view

I'm a student of Physics and I've been studying manifolds and calculus on such objects for a time. Usually when we deal with vector calculus there are books that bring one operational point of view. For example: the book Mathematical Methods for Physicists by George Arfken.

This book brings interpretations of all the objects, like the vectors themselves, the integrals, the operations and so on and in the same time shows how one operates with them in practice. How to manipulate those objects and even carry down computations with them.

Calculus on manifolds, on the other hand, is being a little more complicated. The reason is that all books I've found until now focus just on theorems and their proofs. There's nothing wrong with it, of course this is interesting as well, but what I really need is that operational view with interpretations and so on.

For what I've seem until now, when doing calculus on manifolds one gets one incredible amount of work just to do some computations: work out charts, prove they are bijections and homeomorphism, prove they are $C^k$ related and so on. This also confuses me, because it seems much more complicated than vector calculus even to get started, while many people say it's not.

So, where can I learn this operational point of view of calculus on manifolds? Meaning, learn to interpret objects like exterior derivatives, differential forms and their integrals, and in the same time learn how to in practice carry out operations with those objects?

The basic point of manifold theory, the point which we make great effort to affirm in texts and elementary treatments, is that when you do calculus on a manifold it is the same as it is in $\mathbb{R}^n$ locally. If you write the formula for functions from one manifold to another in terms of coordinate charts then the manner in which you differentiate is just the same as you would expect. In short, a manifold is just a space in which we can "do" calculus locally. A few intuitive points on top of this:

1. push-forwards are essentially the same as the problem of converting a PDE from one coordinate system to another. Of course, this is a specialization which assumes the same dimension in domain and codomain, but it serves to give some familarity.

2. pull-back are essentially just subsitution again, for example if $F(x,y) = (x^2+y^2,y)=(u,v)$ then the pull-back of $du \wedge dv$ is simply obtained by plugging in $u=x^2+y^2$ and $v=y$ and proceeding by the formal total differentiation $du = 2xdx+2ydy$ and $dv=dy$ so $F^*(du \wedge dv) = (2xdy+2ydy) \wedge dy = 2x dx \wedge dy$.

3. integration of a differential form on $\mathbb{R}^n$ really amounts to dropping the wedges and integrating as usual, of course, the algebra of the forms implicits an orientation of the integral which is absent in the un-formed language. The orientation is somehow wrapped up in the coordinate charts and is transferred to the $\mathbb{R}^n$ calculations through the pull-backs needed to define the manifold integration.

4. perhaps a problem which you can appreciate more the need for the formalism (and I gather from the careful wording of your question that you are not uninterested in pure mathematics) is the problem of differential equations. Take for a simple example $\frac{dy}{dx}=\frac{-x}{y}$ note that the differential equation is not defined for $y=0$ and yet the solutions $x^2+y^2=R^2$ seem naturally to include such points. Indeed, if we instead look at the differential equation as $\frac{dx}{dy} = \frac{-y}{x}$ then $x=0$ is troublesome. Yet, intuitively, we can see both cases are just coordinate defects. Really the problem is that we are using a square peg for a round hole. The circle is better described as $\frac{dr}{dt}=0$ hence $r = R$ in polar coordinates. Two questions stand out: first, what is the proper object to capture solutions, second, what is the proper object to capture differential equations? The best answer I've seen is given by the so-called EDS theory. Roughly, the differential equation is a differential form and the solution is a submanifold. This gives us a coordinate invariant manner to see differential equations.

As I continue to study manifolds, I'm always looking for better answers to your question. I like you started with a desire to learn how to "do" calculus on a manifold (at the time for the sake of calculating GR) and I agree it takes time to cull these ideas from the standard treatments. Of course, physics texts suffer the opposite problem. One general advice, you probably already know, search for posts about intuition in math. These contain some of what you seek.

• Thanks very much for your answer. Your explanation of the pullback made differential forms much clearer for me. One thing I also fail to see is how to really integrate on manifolds. Books define this with partitions of unity, but I think it's not really feasible to use them because it would involve computing sums of series. How is integration considerd in this operational point of view? Thanks again. – user1620696 Mar 8 '14 at 20:42
• @user1620696 Well, I suppose there is probably an interesting example where an infinity of coordinate charts all overlap, but, I think the case that there are just finitely many is more typical. One thing to keep in mind, we do calculations with respect to an atlas, not a maximal atlas. For a compact manifold, there exists a finite subcover so integration is just adding together finitely many integrals in $\mathbb{R}^n$. In physics, I think we tend to integrate over noncompact spaces, but we assume our fields are physically reasonable so said integrals converge. – James S. Cook Mar 9 '14 at 4:12