Mathematics of Physics and Engineering

viii Mathematics of Physics and Engineering

are often left out of an applied mathematics course, and the study of these
equations in a physics course often leaves the mathematical part somewhat
of a mystery. In our exposition, we maintain the full rigor of mathemat-
ics while always presenting the motivation from physics. We do this for
the classical mechanics, electromagnetism, and mechanics of continuous
medium, and introduce the main topics from the modern physics of rel-
ativity, both special and general, and quantum mechanics, topics usually
omitted in conventional books on "Engineering Mathematics."
Another advantage is the possibility of further exploration through prob-
lems, as opposed to standard end-of-section exercises. This book offers a
whole chapter, about 30 pages, worth of problems, and many of those prob-
lems can be a basis of a serious undergraduate research project.
Yet another advantage is the space to look at the historical developments
of the subject. Who invented the cross product? (Gibbs in the 1880s, see
page 3.) Who introduced the notation i for the imaginary unit sf^ll
(Euler in 1777, see page 79.) In the study of mathematics, the fact that
there are actual people behind every formula is often forgotten, unless it is a
course in the history of mathematics. We believe that historical background
material makes the presentation more lively and should not be confined to
specialized history books.
As far as the accuracy of our historical passages, a disclaimer is in order.
According to one story, the Russian mathematician ANDREI NiKOLAEVlCH
KOLMOGOROV (1903-1987) was starting as a history major, but quickly
switched to mathematics after being told that historians require at least
five different proofs for each claim. While we tried to verify the historical
claims in our presentation, we certainly do not have even two independent
proofs for most of them. Our historical comments are only intended to
satisfy, and to ignite, the curiosity of the reader.
An interesting advice for reading this, and any other textbook, comes
from the Russian physicist and Nobel Laureate LEV DAVIDOVICH LANDAU
(1908-1968). Rephrasing what he used to say, if you do not understand a
particular place in the book, read again; if you still do not understand after
five attempts, change your major. Even though we do not intend to force a
change of major on our readers, we realize that some places in the book are
more difficult than others, and understanding those places might require a
significant mental effort on the part of the reader.
While writing the book, we sometimes followed the advice of the German
mathematician CARL GUSTAV JACOB JACOBI (1804-1851), who used to
say: "One should always generalize." Even though we tried to keep abstract