
My favorite set of exercises for advanced
undergraduate and beginning graduate students can be found in the book,
Berkeley Problems in Mathematics,
Springer, third edition, 2004,
by JorgeNuno Silva (Contributor)
and Paulo Ney De Souza. The book contains approximately nine hundred problems
which have appeared in preliminary exams in Berkeley over the last twenty
years. It is an invaluable source of problems and solutions for every
mathematics student who plans to enter a Ph.D. program.



For supplementary reading, I recommend the
book,
Calculus Made Easy
Being a VerySimplest Introduction
to Those Beautiful Methods of Reckoning Which Are Generally Called by
the Terrifying Names of, by Silvanus P. Thompson and Martin Gardner.
This book will help you learn calculus. It will develop your intuition,
You will see that calculus is EASY! You can find more comments about
Calculus Made Easy on
Amazon.com.



For a more advanced level of calculus, my favorite book
is
Principles of Mathematical Analysis by Walter Rudin.
This book is a classic. I learned calculus from this book a long time
ago. Now my students use it to study advanced calculus.
CONTENTS: The Real and Complex Number System, Basic Topology, Numerical
Sequences and Series, Continuity, Differentiation, The RiemannStieltjes
Integral, Sequences and Series of Functions, Some Special Functions,
Functions of Several Variables, Integration of Differential Forms, The
Lebesgue Theory, Bibliography, List of Special Symbols, Index.



This is a very elegant introduction to differential forms and to Stokes' theorem.
I used this book to teach a vector calculus course.



Proofs from THE BOOK, Springer, third edition, 2003,
by Martin Aigler and Gunter M. Ziegler is a
wonderful book inspired by Paul Erdos. It presents a collection of "perfect"
proofs. Everything in this book should be accessible to readers whose
backgrounds include only a modest amount of techniques from undergraduate
mathematics.
CONTENTS: Preface, Number Theory, Geometry, Analysis, Combinatorics,
Graph Theory, About the Illustrations, Index.

