(on a bit of a roll with the PoincarĂ© 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.

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The original post was in the category: poincare_project but I'm still in the process of migrating categories over.