The books are not for me but rather my wife will be returning to school soon and would like to refresh her mathematics knowledge especially, but also her physics. She will be working on an undergraduate degree in engineering. She hasn’t taken a science or math class for years now and wants to make sure she has what she needs when she walks through the door.
Well, what she needs will be dependent, to varying extent, upon which door she is entering. Metaphorically speaking, the door to Caltech will be much harder to open than the door going into Podunk U. What should be needed is a background in calculus, and I will suggest three different books on levels from simple to complex.
By far the best simple presentation of the calculus is an updated version of a book written almost a century ago. Silvanus P. Thompson who, surprisingly, was an electrical engineer, wrote this original gem, and Martin Garder, of Scientific American fame, revised it to reflect some more modern ideas. This book will teach the essentials of calculus to the nontechnical reader, one with no more background than high school algebra. This is a highly recommended text for a beginner or for one who is seeking simplicity, with clarity. Calculus Made Easy, Silvanus P. Thompson, F.R.S and Martin Gardner, St. Martin’s Press.
On an intermediate level I would recommend George B. Thomas’ Calculus and Analytic Geometry, Addison Wesley Publishing Co. This book is now in something like its 9th edition, having seen numerous upgrades over the past decades. Thomas is an excellent old-time author in this field, and I think he enlisted the aid of the second author, Ross Finney, for the more modern-type upgrades. All in all, this is a lovely blend of the theoretical with the practical. Not for one who wants a firm grasp of foundational calculus, but perfect for engineering-type students. Also, I would not hesitate to get one of the older editions that are available for much reduced prices. Even though they do not have the same up-to-date focus, that is offset by Thomas’ care and precision.
On the advanced level, for the physicist or mathematician, I absolutely love the two-volume set Introduction to Calculus and Analysis, Richard Courant and Fritz John, Interscience Publishers (I believe it has now been reprinted by a different publisher. My Interscience edition is from 1965.) This is a great set of books written by masters in the field, just the right combination of historical perspective and fundamentals. When you grasp the origin of certain basic ideas, they then become so much more meaningful in today’s context. Anyone who works his way through these two volumes will have a really firm grasp of the fundamentals of the calculus. But, it requires a lot of work, and is very demanding. A great set of books, nonetheless.
For physics I still love the three-volume Feynman Lectures on Physics, but that may be a bit much as a “refresher” and more like a good one-year college course in physics. There are zillions of “simple introductions” to physics but I never found one that I could really recommend. Surprisingly, there are great nontechnical introductions to more esoteric subjects like special and general relativity, but none that I really like in general physics. For the purpose your wife has, I would suggest the two-volume set Physics for Scientists and Engineers, by Raymond A. Serway and Robert J. Beichner, Saunders College Publishing. This set is simple enough that it can be perused on one level to refresh a lot of basic concepts, and yet detailed enough that it will be useful later for reference in other engineering courses. I do not consider this to be a great book, as is Feynman’s, but it is decent in its presentation and will be useful both now and later.
Good luck to your wife in her studies.