Mathematics of General Relativity - Tensorial Derivatives

Tensorial Derivatives

Before the advent of general relativity, changes in physical processes were generally described by partial derivatives, for example, in describing changes in electromagnetic fields (see Maxwell's equations). Even in special relativity, the partial derivative is still sufficient to describe such changes. However, in general relativity, it is found that derivatives which are also tensors must be used. The derivatives have some common features including that they are derivatives along integral curves of vector fields.

The problem in defining derivatives on manifolds that are not flat is that there is no natural way to compare vectors at different points. An extra structure on a general manifold is required to define derivatives. Below are described two important derivatives that can be defined by imposing an additional structure on the manifold in each case.