Tangent Vectors

Abstract

The central idea of calculus is linear approximation. In order to make sense of calculus on manifolds, we need to introduce the tangent space to a manifold at a point, which we can think of as a sort of “linear model” for the manifold near the point. Motivated by the fact that vectors in ℝⁿ act on smooth functions by taking their directional derivatives, we define a tangent vector to a smooth manifold to be a linear map from the space of smooth functions on the manifold to ℝ that satisfies a certain product rule. After defining tangent vectors, we show how a smooth map between manifolds yields a linear map between tangent spaces, called the differential of the map, and a smooth curve determines a tangent vector at each point, called its velocity. In the final two sections we discuss and compare several other approaches to defining tangent spaces, and give a brief overview of the terminology of category theory, which puts the tangent space and differentials in a larger context.