example, the geometry around a star is described by a metric tensor at each point, so at each point of the spacetime the value of the metric should be given to solve for the paths of material particles. Another example is the values of the electric and magnetic fields (given by the electromagnetic field tensor) and the metric at each point around a charged black hole to determine the motion of a charged particle in such a field. Vector fields are contravariant rank one tensor fields. Important vector fields in relativity include the four-velocity, , which is the coordinate distance travelled per unit of proper time, the four-acceleration and the four-current describing the charge and current densities. Other physically important tensor fields in relativity include the following: The stress–energy tensor , a symmetric rank-two tensor. The electromagnetic field tensor , a rank-two antisymmetric tensor. Although the word 'tensor' refers to an object at a point, it is common practice to refer to tensor fields on a spacetime (or a region of it) as just 'tensors'. At each point of a spacetime on which a metric is defined, the metric can be reduced to the Minkowski form using Sylvester's law of inertia. 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. Affine connections The curvature of a spacetime can be characterised by taking a vector at some point and parallel transporting it along a curve on the spacetime. An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction. By definition, an affine connection is a bilinear map , where is a space of