Why is General Relativity so damned difficult and inaccessible? Because of tensor fields.

So what’s this about tensors? Well, caclulus might be abstruse enough already, but tensor calculus takes abstrusity (?) to a whole new level. A tensor is a multidimensional generalisation of scalars and vectors, to n-dimensions. So, a scalar quantity such as mass, having one dimension, can also be thought of as a tensor of rank zero. A vector quantity such as velocity can be thought of as a rank one tensor. Things become complicated when dealing with more complex quantities, such as elasticity, a rank four tensor in materials science.

Performing calculus with tensors becomes useful when considering quantities with two or more dimensions. The great benefit is that it allows you to treat quantities with as many dimensions as you need with the same notational convenience.

A tensor field is a tensor quantity that varies across all points in space. A magnetic field, denoted B, is a vector field, and so can be treated as a tensor field of rank one. The space in which the field is mapped may be the standard Euclidiean 3-space, but might well be some other space, like gee, how about a 4-dimensional spacetime? And I haven’t even mentioned Riemannian manifolds yet.