Basics

Abstract

This chapter is devoted to the basic features needed for Cartesian tensors: the components of a position vector with respect to a coordinate system, the scalar product of two vectors, the transformation of the components upon a change of the coordinate system. Special emphasis is put on the orthogonal transformation associated with a rotation of the coordinate system. Then tensors of rank $\ell \ge 0$ are defined via the transformation behavior of their components upon a rotation of the coordinate system, scalars and vectors correspond to the special cases $\ell =0$ and $\ell =1$. The importance of tensors of rank $\ell \ge 2$ for physics is pointed out. The parity and time reversal behavior of vectors and tensors are discussed. The differentiation of vectors and tensors with respect to a parameter, in particular the time, is treated.