Mathematical Physics Vol 1

1. Vector algebra

1.1 Introduction - On scalars, vectors and tensors We encounter various phenomena in the space that surrounds us and define the concepts that characterize them in order to describe them. However, it has been noted that different phenomena can, mathematically, be described in the same way, that is, they can be elements of the same set in which certain mathematical rules apply. Quantities such as: length, area, volume, mass, temperature, pressure, or electric charge can be specified by a single number (namely, the number of units of a conveniently chosen measurement scale, such as: 3 m , 0.5 m 2 , 10 ◦ C , 1 bar , 110 V , etc.). These quantities are called scalars . The choice of scale is a matter of agreement and depends on practical problems (practical needs). However, we also encounter (physical) quantities that require more data (parameters) in order to be defined. Examples of such quantities are: movement of a point, speed, acceleration, force, etc. These quantities are characterized by direction and magnitude, and we call them vectors . Finally, there are quantities that require even more parameters in order to be defined. Thus, for example, inertia, which captures the relation between angular velocity and angular momentum for a rigid body, is determined by nine independent data (components). Such quantities, if they follow specific physical laws, are called tensors . In this chapter we will study vectors. However, before we define vectors and relevant operations, we will define the coordinate system, since we will later need it to work with vectors more conveniently. 1.2 Coordinate system In order to determine the position of geometric objects, it is necessary to define the reference system in relation to which they are observed. The basic idea (Descartes) 1 is to assign a unique n-tuple of numbers to each point in the 1 René Descartes (Latin name Renatus Cartesius) (1596-1650), French philosopher and mathematician. He introduced analytical geometry. His seminal work Géométrie appeared in 1637, as an addition to his work Discours de la méthode .