Vectors, Tensors and Their Representation

Abstract

In mathematics, a vector is defined as an element of a vector space, and a vector space is a commutative (Abelian) group with a scalar multiplication. This is an abstract definition which has many possible realisations (numbers, functions, geometric objects and so on). For our purposes, it is sufficient to consider one of them, namely the geometric object of an arrow in the three-dimensional, Euclidian, physical space ε. Therefore, in our sense a vector a ∈ ε is an arrow which is characterised by a length and a direction. Physical quantities which can be described by such vectors are, for instance, velocity, acceleration, momentum and force. By contrast, scalars are simple numbers and characterise physical quantities without a direction, like mass, density, temperature etc.