Dear Mr Singleton and Mr. Feynman,
Let me apologize for implying that not only I, but also others here may not be highly qualified in every area we comment on. I only know that to be true of myself, but I confess that in my arrogance, especially late at night and after wine, I have implied questions as to some other people's reliability as well. Mr Feynman is quite right to challenge me on this.
I confess too I do not even know what a second master of theoretical physics is, but it is clear Mr. Feynman has a great deal of personal experience and expertise both in theoretical physics and in studying for it. Thus it is no doubt accurate that anyone wanting to follow his footsteps is well advised to learn differential geometry at some point.
To Mr. Feynman, in spite of my rudeness in the way I suggested it originally, I wonder if perhaps you/we could be of even more service to Mr Singleton by proposing something like a timeline for studying these topics. My worry was that we might discourage someone who is now at the Calculus Made Easy stage, by loading on too much too soon.
Of course I admit that a teacher who really understands it can probably make differential manifolds seem natural even to a beginning calculus student.
As to my own specialty I like geometry and calculus, and algebraic geometry, and complex analysis of one and several variables, and algebraic topology, but am far from a master of any of them. I am thoroughly ignorant of physics, but very impressed by the wonderful intuition physicists seem able to bring to bear even on apparently purely mathematical questions, such as the recent breakthroughs in enumerative algebraic geometry wrought by the theory of quantum gravity.
Please feel free to modify for Mr Singleton's benefit the following elementary remarks which are solely from my own perspective:
As to how the various topics tie in,
1) (differential) calculus is usually now regarded as a method of approximating non linear phenomena by linear phenomena, the simplest example being the approximation of a curve, (the graph of a function), near a given point, by its tangent line (the graph of its derivative).
In two variables one approximates a surface, (the graph of a function of two variables), near a given point, by a tangent plane (again the plane is the graph of the derivative, which is a linear function of two variables).
In higher dimensions, not being able to picture the situation so easily, we again approximate non linear phenomena, by things we call "linear" phenomena.
This time we use an algebraic definition of "linearity" to free us from the need to visualize it. I.e. a function L is linear if it preserves addition and scaling, which means that L(x+y) = L(x) + L(y), and L(cx) = cL(x). This definition, and an appropriate definition of "approximates" allows us to do calculus, i.e. linear approximation, in arbitrary dimensions, and even in infinite dimensions.
2) Obviously, since calculus is the science of approximating non linear phenomena by linear ones, it is essential to understand first linear phenomena. This is why you must study linear algebra, essentially as a precursor to higher calculus. In low dimensions calculus is taught without pointing out this connection, because the linear phenomena are so simple as not to need special study, or because some of us were born before the linear algebra revolution, and simply teach the way we learned.
Clearly one cannot even define linearity unless the space on which the functions are defined has a linear structure, i.e. unless you can add x+y in it, so linear algebra is necessarily carried out on linear or flat spaces.
3) Unfortunately, or interestingly as it may be, space time is not flat according to Einstein who postulated that mass causes it to be curved (here I am on thin ice, and out of my depth, so please help me out here Feynman). Thus how can we use calculus to study general relativity? or space time at all?
The answer is to note that it is not the original non linear function that must be defined on a linear space but only its derivative, i.e. its linear approximation. So here come the idea of differential manifolds, a differential manifold is a curved space, like a curved surface on which one has defined non linear phenomena of interest, and then one proceeds in two stages to approximate it by linear phenomena.
First one approximates the curved space itself by a flat space, which is again called the tangent space, and then one approximates the original non linear function which was defined on the curved space, by a linear object which is defined on the tangent space.
For instance if we regard the surface of the Earth as a sphere, the sphere is the curved space, and we could be interested in the phenomenon of motion of water on the surface of the earth. To approximate this linearly, we fix a given point at which to make our study, say the north pole, and approximate the sphere there by a flat tangent plane. Better yet we put a (different) tangent plane at every point of the sphere.
Then we approximate the flow of water on the sphere by assigning at each point, a velocity vector in the direction of the flow, which vector lies in the tangent plane at the given point. This "vector field" is analogous to the derivative of a function in calculus.
4) The theory above is called differential manifolds, but the theory of differential geometry goes one step further and adds a notion of length of tangent vectors, usually called a "metric" or Riemannian metric on the tangent bundle as Mr Feynman correctly said. This enables one to compare the extent of curvature at different points of our space, hence to distinguish a large sphere from a small one, such as a small moon around our planet.
I am not too up on differential geometry, but the key concept seems to be curvature which is measured by an object called a "tensor" which comes from "multilinear algebra" (the second course in linear algebra). Basically the "first" derivative is a linear object and the "second" (or higher) derivatives are bilinear (or multilinear) objects.
(Basically, linear and multiplinear mathematics are easier than non linear mathematics, and so the main job in many areas of math is merely to approximate non linear objects by linear ones, or by families of linear ones, and so one simply must study as much linear maths as possible.)
5) Further, not only is space not flat, but it is not as simple as a sphere either, possibly taking more exotic shapes like the surface of a doughnut or a three dimensional analog of the surface of a doughnut, with many (possibly "black") holes of varying dimensions. Then one wants to study these holes, or "connectivity" phenomena as well. This subject is called algebraic or differential topology. I.e. the subject is topology, and one studies it by various tools available, algebra or calculus.
Now as to sources for studying these things, if you already know some calculus you will learn little or nothing of value, except some humor, by reading either Calculus Made Easy or Street Wise Calculus, (and I admit that I object on intellectual grounds to books such as the latter).
The first and still most rigorous and advanced introduction to calculus with linear algebra is the book Foundations of Modern Analysis by Dieudonne, from which sophomore (!) honors calculus used to be taught at Harvard in the 1960's. This book covers metric spaces, Hilbert space, calculus on Banach spaces, one complex variable, and some differential equations including Sturm Liouville theory, but nothing on manifolds. There are no figures in it, as Dieudonne was so strict a mental master as to disbelieve in allowing illustrations to learners. Fortunately his example is not much followed.
An easier source for learning is the book by Marsden and Tromba, written since then as an attempt to render this topic genuinely accessible to good college students, and used not too long ago at Berkeley. (Marsden was also a calculus instructor, and very well liked, at Harvard in the 60's). I have even taught from this book to high school students, one of whom is now a math prof (in topology) at a major university. It begins with a review of linear algebra, which Dieudonne assumes. It is also restricted to finite dimensions.
For a short succint introduction to "calculus on manifolds", it is hard to beat the book of that title by Michael Spivak. Spivak also has written the definitive textbook on differential geometry of our generation, in 5 volumes! available from his website at Publish or Perish. He is also a nice guy, brilliant, and a wonderful teacher. He himself was a differential topologist, with a construction in that subject named for him called the "Spivak normal fibre bundle", and he studied with both the greats John Milnor and Raoul Bott.
Algebraic topology can be hard to learn in my experience, but there is a fantastic book on differential topology by Guillemin and Pollack, a skillful rewrite for undergraduates of an even better but more advanced short book by John Milnor.
Spivak's differential geometry book also contains a wonderful section in volume one(?) on deRham cohomology, or algebraic topology taught from the perspective of calculus. Guillemin and Pollack also treat some deRham cohomology, which is basically path integration, largely borrowed from Spivak, at the end of their book, and they do a very nice job.
So may I suggest you read first Marsden and Tromba, then Guillemin and Pollack or Spivak's little book. Then maybe Spivak's first or second volume of differential geometry, and then take stock of your situation.
You might also try (for algebraic topology) the introduction by William Fulton, an algebraic geometer, who makes connections with other subjects in it nicely.
But by now someone else should chime in.
best regards,
roy
