8 1 Introduction
1.3 Remarks on History and Literature
Cartesiantensors, as they are used here for the description of material properties in
anisotropic media, were first introduced by the German physicistWoldemar Voigt
around 1890. This is documented in his books on the physics of crystals [1]. In his
lectures on theoretical physics, published around 1895, Voigt worked with tensors,
without using the word. In the books on crystal physics, the termvectoris used as
if everybody is familiar with it. Tensors, in particular of rank two, are discussed in
detail. Tensors of rank three and four are applied for the description of the relevant
physical phenomena. Voigt refers to Pierre Curie [2] as having very similar ideas
about tensors and symmetries.
In connection with differential geometry, the notiontensor, however, not the
word, was already invented byCarl Friedrich Gauss, who had lived and worked in
Göttingen, more than half a century before Voigt. At about the same time as Voigt, the
Italian mathematicians Tullio Levi-Civita and Gregorio Ricci Curbastro formulated
tensor calculus, as it is used since then in connection with differential geometry [3].
Their work provided the mathematical foundation for Einstein’sGeneral Relativity
Theory[4]. This topic, however, is not treated here. The necessary mathematical tool
for General Relativity are found in the text books devoted to this subject, e.g. in
[5–7].
As stated before, the emphasis of this book is on Cartesian tensors in three dimen-
sions and applications to physics, in particular for the description of anisotropic
properties of matter. Classic books on the subject were published between 1930 and
1960, by Jeffrey [8], Brillouin [9], Dusschek and Hochrainer [10], and Temple [11].
In the Kinetic Theory of gases, tensors and the importance of the use of irreducible
tensors was stressed in the book of Chapman and Cowling [12] and in theHandbuch
article by Waldmann [13]. I was introduced to Cartesian tensors in lectures on elec-
trodynamics by Ludwig Waldmann in 1963. Applications to the kinetic theory of
molecular gases, in the presence of external fields, as well as to optics and transport
properties of liquid crystals required efficient use of tensor algebra and tensor calcu-
lus. This strongly influenced a book for an introductory course to vectors and tensors
[14], for first year students of physics, and led to a collection of computational rules
and formulas needed in more advanced theory [15]. For the application of tensors in
the kinetic theory of molecular gases, see also [16, 17].
In the following, no references will be given to the physics which is standard in
undergraduate and graduate courses, the reader may consult her favorite text book or
internet source. In the second part of the book, devoted to more specialized subjects,
references to original articles will be presented. The particular choice of these topics
largely reflects my own research experience.
A remark on vectors is in order. The notion ofvector, i.e. that some physical
quantities, like velocity and force, have both a magnitude or strength and a direction
and that the combined effect of two vectors follow a rule which we call theaddition
of vectors, was known long before the wordvectorwas used. Among others, Isaac
Newton was well aware, how vectorial quantities should be handled. Gauss and others