Covariant derivative

Summary

A covariant derivative differentiates a vector field in the direction of a tangent vector, and the result is again a tangent vector. It applies more generally to sections of a vector bundle and to tensor fields. It is the operator $\nabla_X$ that a connection provides. Unlike partial derivatives of components, it does not depend on the choice of coordinates. It also differentiates vector fields along curves, which leads to parallel transport and geodesics.