Why did I write this book? I am frequently struck by the fact that students
with an undergraduate degree in mathematics know so little about the in-
terrelations between various parts of mathematics. Even those at the start
of graduate school so suffer. They may know a lot of algebra and analysis,
for example, but few of them have a clue that there are bridges that connect
these two subjects and that each affects the other. This book is an attempt
to address this situation.
Having a broad view of mathematics is an advantage whether you are a
high school teacher, an industrial practitioner, a professor at a liberal arts
college, or a research mathematician. Everyone, including the author, has
much to learn about the interconnections between various parts of mathe-
matics; though those engaged in exploring the boundaries of mathematics
seem to eventually discover several of these connections, at least those as-
sociated with their own research. I still remember my delight, as a young
assistant professor, at discovering the true test for the nature of a critical
point of a function of two variables. That delight turned to a feeling that my
education was at fault and then to the realization that I was one of many
in the same boat going upstream in the river of mathematics.
This book presents material for a senior-level course for mathematics
majors, including those who intend to become school teachers. Most chap-
ters explore relations between different parts of mathematics. The chapters
are reasonably self-contained, but some require more sophistication than
others. In fact anyone who examines this book will discover that it is far
from homogeneous either in its content or in its demands on the reader in
both background and effort. That’s intentional and is essentially required by
the the great variation in undergraduate preparation.
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