Chapter 1
Introduction
Abstract In this chapter, preliminary remarks are made on vectors and tensors.
The axioms of a vector space and the norm of a vector are introduced, the role of
vectors for classical physics and for Special Relativity is discussed. The scope of the
book as well as a brief overview of the history and literature devoted to vectors and
tensors are presented. Before tensors and their properties are introduced here, it is
appropriate to discuss the question: what is a vector? As we shall see, mathematicians
and physicists give somewhat different answers.
1.1 Preliminary Remarks on Vectors.
Some physical quantities like themass, energyortemperatureare quantified by a
single numerical value. Such a quantity is referred to asscalar. For other physical
quantities, like thevelocityor theforcenot only their magnitude but also their direc-
tion has to be specified. Such a quantity is avector. In the three-dimensional space
we live in, three numerical values are needed to quantify a vector. These numbers
are, e.g. the three components in a rectangular, Cartesian coordinate system.
In general terms, a vector is an element of a vector space. The axioms obeyed by
these elements are patterned after the rules for the addition of arrows and for their
multiplication by real numbers.
1.1.1 Vector Space
Consider special vectors, represented by arrows, which have a length and a direction.
The rules for computations with vectors can be visualized by manipulations with
arrows. Multiplication of a vector by a number means: the length of the arrow is
multiplied by this number. The relation most typical for vectors is the addition of
two vectorsaandbas indicated in Fig.1.1.
The operationa+bmeans: attach the tail ofbto the arrowhead ofa.Thesumis
the arrow pointing from the tail ofato the arrowhead ofb.Thesumb+a, indicated
by dashed arrows, yields the same result, thus
© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_
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