This book takes the reader on a journey through the world of
college mathematics, focusing on some of the most important
concepts and results in the theories of polynomials, linear
algebra, real analysis, differential equations, coordinate
geometry, trigonometry, elementary number theory, combinatorics,
and probability. Preliminary material provides an overview of
common methods of proof: argument by contradiction, mathematical
induction, pigeonhole principle, ordered sets, and invariants. Each
chapter systematically presents a single subject within which
problems are clustered in each section according to the specific
topic. The exposition is driven by nearly 1300 problems and
examples chosen from numerous sources from around the world; many
original contributions come from the authors. The source, author,
and historical background are cited whenever possible. Complete
solutions to all problems are given at the end of the book. This
second edition includes new sections on quad ratic polynomials,
curves in the plane, quadratic fields, combinatorics of numbers,
and graph theory, and added problems or theoretical expansion of
sections on polynomials, matrices, abstract algebra, limits of
sequences and functions, derivatives and their applications,
Stokes' theorem, analytical geometry, combinatorial geometry, and
counting strategies. Using the W.L. Putnam Mathematical Competition
for undergraduates as an inspiring symbol to build an appropriate
math background for graduate studies in pure or applied
mathematics, the reader is eased into transitioning from
problem-solving at the high school level to the university and
beyond, that is, to mathematical research. This work may be used as
a study guide for the Putnam exam, as a text for many different
problem-solving courses, and as a source of problems for standard
courses in undergraduate mathematics. Putnam and Beyond is
organized for independent study by undergraduate and gradu ate
students, as well as teachers and researchers in the physical
sciences who wish to expand their mathematical horizons.
General
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