In the previous Poincaré Project post about coordinate systems and metrics, we introduced the notion of a metric that tells us how quickly a coordinate changes on a one-dimensional manifold. Let's now extend that to two (or more) dimensions.
Imagine that you're at a particular point on a two-dimensional manifold. If you head off in a particular direction from that point at a particular rate, your coordinates will change. The metric tells you, from a given point, the rate of change of each of your coordinates given travel in a particular direction at a particular rate.
Or to make it more concrete, imagine you're in a boat on the ocean and you start to travel due east at ten knots. The metric will tell you the rate of change of longitude.
Note three things:
To better understand what kind of mathematical object this metric is, we'll need to better understand the notion of a vector.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.