Metrics in Two or More Dimensions

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:

• the metric is different at different locations. In our ocean example, the metric will be different at different latitudes.
• the "travel in a particular direction at a particular rate" is a kind of vector.
• the metric, at a particular point, is a linear function of that vector. If you head off twice as fast, the coordinates will change twice as fast, and so on.

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