Many colleagues and students in the mathematical community have made valuable contributions and suggestions since this book was begun. An early draft of the book was written in collaboration with Ralph Abraham. We thank him for allowing us to draw upon his work. It is impossible to list all those who assisted with this book, but we wish especially to thank Michael Hoffman and Joanne Seitz for their help on earlier editions. We also received valuable comments from Mary Anderson, John Ball, Patrick Brosnan, Andrea Brose, David Drasin, Gerald Edgar, Michael Fischer, Frank Gerrish, Mohammad Gohmi, Jenny Harrison, Jan Hogendijk, Jan-Jaap Oosterwijk, and Anne van Weerden (Uterecht), David Knudson, Richard Kock, Andrew Lenard, William McCain, Gordon McLean, David Merriell, Jeanette Nelson, Dan Norman, Keith Phillips, Anne Perleman, Oren Walter Rosen, Kenneth Ross, Ray Sachs, Diane Sauvageot, Joel Smoller, Francis Su, Melvyn Tews, Ralph and Bob Tromba, Steve Wan, Alan Weinstein, John Wilker, and Peter Zvengrowski. The students and faculty of Austin Community College deserve a special note of thanks, as do our students at both Caltech and UC Santa Cruz.

We owe a very special thanks to Stefan Hildebrandt and Robert Palais for their historical advice.

We are grateful to the following instructors who have provided detailed reviews of the manuscript: Dr. Michael Barbosu, SUNY Brockport; Brian Bradie, Christopher Newport University; Mike Daven, Mount Saint Mary; Elias Deeba, University of Houston–Downtown; John Feroe, Vassar; David Gurari, Case Western Reserve; Alan Horowitz, Penn State; Rhonda Hughes, Bryn Mawr; Frank Jones, Rice University; Leslie Kay, Virginia Tech; Richard Laugesen, University of Michigan; Namyong Lee, Minnesota State University; Tanya Leiese, Rose Hullman Institute; John Lott, University of Michigan; Gerald Paquin, Universit´e du Qu´ebec à Montr´eal; Joan Rand Moschovakis, Occidental College; A. Shadi Tahvildar-Zadeh, Princeton University; Howard Swann, San Jose State University; Denise Szecsei, Stetson University; Edward Taylor, Wesleyan; and Chaogui Zhang, Case Western Reserve. For the fifth edition, we want to thank all the reviewers, but especially Andrea Brose, UCLA, for her detailed and valuable comments. For the present edition we would like to thank Eliot Brenner, University of Minnesota; Bueno Cachadina Maribel, UCSB; Evan Merrill Bullock, Rice University; Xiaodong Cao, Cornell University; Der-Chen Chang, Georgetown University; Lenny Fukshansky, Claremont McKenna College; Ralph Kaufmann, Purdue University; Mohammed Kazemi, University of North Carolina; Min-Lin Lo, California State University San Bernardino; Douglas Meade, University of South Carolina; Steven Miller, Brown University; Doug Moore, UCSB; Eric J. Moore, University of Toronto Scarborough; Peter Nyikos, University of South Carolina; Olga Radko, UCLA; David Russell, Virginia Tech; Francisco J. Sayas, University of Minnesota; Ryan Scott, Rice University; Shagi Di Shih, University of Wyoming; Joel Spruck, Johns Hopkins University; Graeme Wilkin, Johns Hopkins University; I Wu, Johns Hopkins University. Most important of all are the readers and users of this book whose loyalty for over 35 years has made the sixth edition possible.

A final word of thanks goes to those who helped in the preparation of the manuscript and the production of the book. For the earlier editions, we thank Connie Calica, Nora Lee, Marnie McElhiney, Ruth Suzuki, Ikuko Workman, and Esther Zack for their excellent typing of various versions and revisions of the manuscript; Herb Holden of Gonzaga University and Jerry Kazdan of the University of Pennsylvania for suggesting and preparing early versions of the computer-generated figures; Jerry Lyons and Holly Hodder for their roles as our previous mathematics editors; Christine Hastings for editorial supervision; and Trumbull Rogers for his expert copyediting.

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For the sixth edition we would like to acknowledge the strong support of our acquisitions editor Terri Ward, our project editor Katrina Wilhelm, and once again Vivien Weiss, our copy and production editor, as well as Brittney Corrigan-McElroy, without all of whose dedicated and talented efforts this new edition would not have been possible. The addition of so many excellent new exercises was made possible by the contributions of Corey Shanbrom and Paul Tokorcheck, both of whom have extensive experience teaching out of this text.

Finally, the current edition would have been impossible without the tireless efforts and total dedication of Wendy McKay and Nafiseh Khoram as well as the generous support of the California Institute of Technology. Jerry Marsden was particularly proud to be associated with this outstanding institution.

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