1.3 Remarks on History and Literature 9
used a geometric representation of complex numbers in a way which was essentially
equivalent to dealing with two-dimensional vectors. William Rowan Hamilton tried
to extend this concept to three dimensions. He did not succeed but, in 1843, he
invented the four-dimensionalquaternions. James Clark Maxwell did not encourage
the application of quaternions to the theory of electrodynamics, but rather favored a
vectorial description. Vector analysis, as it is used nowadays, was strongly promoted,
around 1880, by Willard Gibbs [18].
1.4 Scope of the Book
The first part of the book, Chaps. 2 – 8 , is a primer on vectors and tensors, it provides
definitions, rules for calculations and applications every student of physics should
becomefamiliarwithatanundergraduatelevel.Thesymmetryofsecondranktensors,
viz. their decomposition into isotropic, antisymmetric and symmetric traceless parts,
the connection of the antisymmetric part with a dual vector, in 3D, as well as the
differentiation and integration of vector and tensor fields, including generalizations
of the laws of Stokes and Gauss play a central role.
Part II, Chaps. 9 – 18 , deals with more advanced topics. In particular, Chaps.
9 – 11 are devoted to irreducible tensors of rank, multipole potentials and multipole
moments, isotropic tensors. Integral formulae and distribution functions, spin oper-
ators and the active rotation of tensors are presented in Chaps. 12 – 14. The properties
of liquid crystals intimately linked with tensors, constitutive relations for elasticity,
viscosity and flow birefringence, as well as the dynamics of tensors obeying nonlin-
ear differential equations are treated in Chaps. 15 – 17. Whereas this book is mostly
devoted to tensors in 3D, Chap. 18 provides an outlook to the 4D formulation of
electrodynamics. Answers and solutions to the exercises are given at the end of the
book.
The examples presented are meant to show the applications of tensors in a variety
of physical properties and phenomena, occurring in different branches of physics.
The examples are far from exhaustive. They are closely linked with the author’s expe-
rience in teaching and research. Applications toMechanicsand toElectrodynamics
and Opticsare, e.g., found in Chaps. 2 – 10 and in Chaps. 7 – 14 , as well as in Chap. 18.
Applications toQuantum Mechanicsand properties ofAtoms and Moleculesare dis-
cussed in Chaps. 5 , 7 , and 10 – 13 .Elasticity, HydrodynamicsandRheologyare treated
in Chaps. 7 – 10 and 16. Problems ofStatistical Physics,ofthePhysics of Condensed
MatterandMaterial Propertiesare addressed in Chaps. 12 – 16. Applications toNon-
equilibrium PhenomenalikeTransport and Relaxation ProcessesandIrreversible
Thermodynamicsare presented in Chaps. 12 , 14 , 16 , and 17. The physics underlying
the various applications of tensors is discussed to an extend considered appropriate,
without the intention to replace any textbook or monograph on the topics considered.