What is a tangent space, anyway?

My site is called the Tangent Space. This is something of a pun, on the fact I am prone to tangents and parentheticals. But actually, the tangent space is a concept in mathematics. Tangent spaces come up in both differential and algebraic geometry. They’re easier to understand in differential geometry though, so that will be the focus of this post.[1] There are actually several ways to define tangent spaces, that are best suited to different contexts (and in “good” situations, the different definitions are compatible). Differential geometry studies structures called manifolds. A manifold is some kind of geometric object in which near every point there is a local coordinate system. Also, we should know how to transition between the coordinates for one region and another when they overlap. Each coordinate system is called a chart, and the collection of charts covering the whole manifold is called an atlas. A good first example of a manifold is the surface of a sphere — for example, the…