I have been rereading the first part of Baez and Muniain, on reformulating electromagnetism in the language of differential geometry.  Here are some notes; they mostly follow the book, but only the parts necessary for writing down and understanding the final equations.

Manifolds

   The spaces physicists are interested in studying are locally similar to \mathbb{R}^n.  For example, the 2-sphere S^2 – i.e., the surface x^2 + y^2 + z^2 = 1 - is locally similar to the plane \mathbb{R}^2, which is why the world looks flat.  To generalize this idea to spaces that look like \mathbb{R}^n, we define n-dimensional manifolds.

   First, we must define topological spaces.  A topological space is a set X and a collection of subsets of X, called the open sets, which satisfy: 1) the null set and X are open, 2) the intersection of any two open sets is also open, and 3) the union of any number of open sets is also open.

   Topological spaces allow the definition of continuous functions.  A function between two topological spaces, f\colon X \rightarrow Y, is continuous if given any open set U \subseteq Y in the target space, the inverse image f^{-1}U \subseteq X is open.  Basically, this means the function sends “nearby” elements of X - “nearby” meaning the elements are all members of some open set – to “nearby” elements of Y.

   Now, we can see how to use the idea of open sets to link topological spaces and \mathbb{R}^n.  Consider a topological space M, covered with open sets U_i.  For each open set, we define a continuous function, called a chart: \varphi_i\colon U_i \rightarrow \mathbb{R}^n.  If these charts are defined such that the transition function \varphi_i \circ \varphi^{-1}_j\colon \mathbb{R}^n \rightarrow \mathbb{R}^n between the two \mathbb{R}^n spaces associated with the open sets U_i and U_j is smooth, then M is called a smooth n-dimensional manifold.  This manifold is a space of elements, separated into “nearby” groups, each of which can be related to \mathbb{R}^n; thus, the manifold looks like smoothly-connected patches of n-dimensional Euclidean space.  Using these local Euclidean spaces, we can define smooth functions on the manifold in a familar manner; e.g., f\colon M \rightarrow \mathbb{R} is smooth if f \circ \varphi^{-1}_i\colon \mathbb{R}^n \rightarrow \mathbb{R} is smooth for all i.  We can then use these functions to define vector fields, and more, on the manifold, as if we were working in \mathbb{R}^n as usual.

Vector Fields

   We are used to thinking of a vector v \in \mathbb{R}^n as a coordinate n-tuple, with n components v^{\mu}: (v^1,...,v^n).  We multiply these components by the “basis vectors” and add everything up to get the vector “object.”  However, this picture is unclear for manifolds; how should one define simultaneous basis vectors for two vectors at different points on the sphere, for example?  If one thinks of these two vectors as little arrows tangent to the sphere, they might lie in different planes entirely, which makes defining basis vectors and coordinates difficult.  Thus, we would like a coordinate-free definition of vector fields.

   To start, we note that one can differentiate a function on the manifold in the direction of a vector, with the directional derivative.  Consider the usual directional derivative of f in the direction of v, which we shall write as vf; this is simply \nabla f = v^{\mu}\partial_{\mu}f.  Let us identify the vector field v not with the components v^{\mu}, but instead with the operator v^{\mu}\partial_{\mu}.  What does this mean?  Given a vector field v=v^{\mu}\partial_{\mu} and a function f on the manifold, at each “point” of the manifold we’ll take the derivative of f in the direction of v, giving us a new function v^{\mu}\partial_{\mu}f.  So combining a vector field with a function gives us another function on the manifold, which is related to derivatives of the original function.

   Now we can generalize this new picture of a vector field to a coordinate-free abstraction that contains all of the essential properties of the directional derivative.  Let C^{\infty}(M) be the set of real-valued smooth functions (which have infinitely many continuous derivatives) on the manifold M.  C^{\infty}(M) is a commutative algebra over \mathbb{R}, which just means that elements of C^{\infty}(M) combined with elements of \mathbb{R} obey a number of addition, multiplication, and distributive properties.  We define a vector field v on M to be a function v\colon C^{\infty}(M) \rightarrow C^{\infty}(M), obeying: 1) linearity over C^{\infty}(M), and 2) the Liebniz law v(fg) = fv(g)+v(f)g, where f,g \in C^{\infty}(M).  Let Vect(M) be the set of all vector fields on M.  It is easy to check that Vect(M) is a vector space over C^{\infty}(M).  Thus, this abstract definition satsifies our usual ideas of a vector field, without any coordinates involved.

   Finally, we can define “arrows” at each “point” of the manifold, as in our original picture of the sphere and arrows tangent to points on it.  We will just evaluate the function returned by the vector field at the relevant point p \in M, i.e. v(f)(p).  Defining the tangent vector v_p\colon C^{\infty}(M) \rightarrow \mathbb{R} by v_p(f) = v(f)(p), we think of the real number v(f)(p) as the result of differentiating the function f in the direction of the tangent vector v_p.  We define the tangent space at p, T_pM, to be the set of all v_p.

   We can see that our original picture of identifying vectors with their components stems from the nature of the tangent space of \mathbb{R}^n.  Since for any point p \in \mathbb{R}^n, the tangent vectors (\partial_{\mu})_p \in T_p\mathbb{R}^n form a basis, we sloppily identify two vectors v_p and v_q at different points to be the same if v_p^{\mu}=v_q^{\mu}.  However, in actuality v_p=v_p^{\mu}(\partial_{\mu})_p \in T_p\mathbb{R}^n and v_q=v_q^{\mu}(\partial_{\mu})_q \in T_q\mathbb{R}^n are vectors in different tangent spaces, even if v_p^{\mu}=v_q^{\mu}.  We can get away with this sloppiness in \mathbb{R}^n, but not when dealing with more complex manifolds.

   In any case, this coordinate-free definition of vector fields allows us to define even more structures on the manifold.  These will be useful in distilling the essential qualities of Maxwell’s Equations.