Preface

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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