Manifolds, Tensors, and Forms (An Introduction for Mathematicians and Physicists) – 1st Edition

About the Author

Paul Renteln is Professor of Physics in the Department of Physics, California State University, San Bernardino, where he has taught a wide range of courses in physics. He is also Visiting Associate in the Department of Mathematics, California Institute of Technology, where he conducts research into combinatorics.

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Product details


Cambridge University Press






1 edition
(November 21, 2013)

Page Count


About the Author

Paul Renteln


This book offers a concise overview of some of the main topics in differential geometry and topology and is suitable for upper-level undergraduates and beginning graduate students in mathematics and the sciences. It evolved from a set of lecture notes on these topics given to senior-year students in physics based on the marvelous little book by Flanders, whose stylistic and substantive imprint can be recognized throughout. The other primary sources used are listed in the references.
By intent the book is akin to a whirlwind tour of many mathematical countries, passing many treasures along the way and only stopping to admire a few in detail. Like any good tour, it supplies all the essentials needed for individual exploration after the tour is over. But, unlike many tours, it also provides language instruction. Not surprisingly, most books on differential geometry are written by mathematicians. This one is written by a mathematically inclined physicist, one who has lived and worked on both sides of the linguistic and formalistic divide that often separates pure and applied mathematics. It is this language barrier that often causes
the beginner so much trouble when approaching the subject for the first time. Consequently, the book has been written with a conscious attempt to explain as much as possible from both a “high brow” and a “low brow” viewpoint,1 particularly in the early chapters.
For many mathematicians, mathematics is the art of avoiding computation. Similarly, physicists will often say that you should never begin a computation unless
you know what the answer will be. This may be so, but, more often than not, what happens is that a person works out the answer by ugly computation, and then reworks and publishes the answer in a way that hides all the gory details and makes it seem as though he or she knew the answer all along from pure abstract thought. Still, it is true that there are times when an answer can be obtained much more easily by means of a powerful abstract tool. For this reason, both approaches are given their due here. The result is a compromise between highly theoretical approaches and concrete calculational tools.