(on a bit of a roll with the Poincaré Project recently)
Back in 2006, I talked about the notion of a coordinate system which, for some region of a manifold, provides a tuple of real numbers identifying each point in that region. The mapping is homeomorphic so the coordinate system is continuous.
You can think of each component of the tuple as a separate function that maps from a point on the manifold to a real number. So, for a two-dimensional manifold, there's an x-coordinate function and a y-coordinate function that respectively tell you the x-coordinate and y-coordinate of a given point. The tuple for point p is then just (x(p), y(p)). The situation for a three-dimensional manifold is similar, you just need a third coordinate function. (Note these functions aren't inherent to the manifold, they are structure added to a manifold and you can, of course, have any number of alternative coordinate systems for the same region of a manifold.)
Each of these coordinate functions is no different than something like temperature or electric potential. They are all scalar fields—continuous real-valued functions on a region of a manifold—that give a real value for every point in that region in a way that if you traveled along a continuous path through the manifold, the value would change continuously.
Now say we wanted to describe how one of these coordinate functions changes as one moved in a particular direction from a particular point on the manifold. Well, we've already seen that
stack-type vectors are about rates of change of some quantity as position changes.
so you can actually think of a coordinate system as defining N stack-type vectors at every point, where N is the number of dimensions of the manifold. Just look at a piece of graph paper and you can see the stack-type vectors. Because the stack-type vectors are showing the rate of change of the coordinate, they are actually the derivative of the coordinate with respect to position (i.e. the gradient).
You can probably already see it, but something interesting happens when we combine the notion of traveling in a particular direction at a particular rate with the notion of a particular quantity (coordinate or otherwise) changing according to position. We'll talk about that next.
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.