vi Preface
as carefully as an alert physicist reads an accountof a newly developed
nuclear reaction, and to learn as much as he can. In most cases a reason-
able investment of time can produce satisfactory understanding of the
text as well as solutions of several of the problems at theend of the sec-
tion. Thus average students can make satisfactory progress. In some
cases it is an almost superhuman task todigest all of the problems and
remarks at the end of a section before additional mathematics has been
studied. Thus superior students have ample opportunities to acquire
large amounts of additional information and skill.
To a considerable extent, this book is a book about mathematics as
well as a mathematics textbook that teaches formulas and procedures.
The historical and philosophical aspects of our subject are not neglected.
The text, problems, and remarks frequently give students quite unusual
opportunities and incentives to think and to become genuine authorities
on developments of ideas, terminologies, notations, and theories. The
book strives to produce thoughtful articulate and perhaps even some-
what sophisticated students who will find that their course in calculusgives them admirable preparation for intellectual pursuits. It is fre-
quently said that calculus textbooks contain so little of the spirit and con-
tent of modern mathematics that they do not enable students to decide
whether they have the interests and the aptitudes required for life-long
careers in pure mathematics or in another science in which mathematics
plays a major role. Hopefully, this book does.
The first third of the book contains all or nearly all of the information
about analytic geometry, vectors, and calculus that students normally
need in their introductory full-year college and university courses in
physics. One distinguishing feature of the book is the early introduc-
tion and continued use of vectors in three-dimensional space. These vec-
tors simplify, clarify, and modernize our mathematics and, at the same
time, make our course more interesting to teachers and vastly more inter-
esting and immediately useful to students. Modern meaningful defi-
nitions and terminologies of the calculus are used, but we retain and
explain the standard notations so students can be prepared to live in the
parts of the world outside their own calculus classrooms.
The logical structure of the book should be explained. We make no
effort to tell what points, lines, and planes are; we suppose that they
exist, and use the axioms of the geometry of Euclid. Similarly we make
no effort to tell what real numbers are; we suppose that the things exist
and use the axioms that govern operations involving them. The book
is based upon these axioms. If a theorem fails to have enough hypotheses
to imply the conclusion, it is a blunder. If an assertion or definition is
meaningless, it is a blunder. If an argument purported to be a proof or
a derivation has a flaw, it is a blunder. If we pretend to prove a formula
for something that has not been defined, this is a blunder. Being