Differential Equations with Applications and Historical Notes, George F. Simmons, McGraw-Hill, 1972.

I acquired this from Amazon based on a recommendation from Steven Strogatz's (@stevenstrogatz) twitter feed on January 9.  Strogatz commented: My favorite preface to a math book is that written by George Simmons in "Differential Equations with Applications and Historical Notes." He then went to on to quote part of the preface.  It was so attractive that I had to see the book, so I ordered a copy.  I got a first edition copy, there are also second and third editions.  The book is well suited to physic and engineering students.  I could say more, but check it out in your library - it is well worth your time.

Introductory Eigenphysics, Clive A. Croxton, John Wiley & Sons, London, 1974

I recently acquired this is as a used book on Amazon, the forward was written in 1973, two years before I started university.  I wish I had found it long ago.  It is a unified treatment of the partial differential equations of physics.  Croxton's stated aim is to "attempt to break down the conventional divisions between quantum and classical mechanics, between physical optics and electron scattering, between electrostatics and aerodynamics, and so on . . .."  The author succeeds admirably.  If you have never seen the book, take a look at it unifies material across multiple different courses.  My copy was a withdrawn copy from the Cal Poly library.

The availability of this book begs the question: why did a library retire a book of enduring interest?  It is useful today as the day it was written.  Many years ago when the library at the University of Central Arkansas was culling its collection, I was the library representative for the Department of Physics.  They wanted to retire many books simply because they were old.  I only allowed them to cull some books they shouldn't have acquired in the first place (think vanity press publications such as a book on general relativity by a urologist as a typical example.)

Anyway, this book is a hidden gem - every bit as good as his famous text on liquid state physics.

How to Think about Analysis, Lara Alcock, Oxford University Press, Oxford, 2014

I have previously commented on Alcock's earlier work, How to study as a Mathematics Major.  This book is even better.  The first 50 or so pages are about learning analysis. The balance of the book reviews the central concepts of analysis unencumbered by the need to prove everything - which will, of course, be done in the actual course.  So what we have here are research-informed strategies for learning analysis.  I would recommend this to all students (Math, Physics, Chemistry, EE, etc.) who need to take advanced courses in mathematics.  It would also help faculty who are preparing mathematically oriented courses that make use of the material covered in rigorous calculus and real analysis classes.

Advanced Calculus for Applications, Francis B. Hildebrand, Prentice-Hall, Englewood Cliffs, NJ, 1962

In an earlier review, I promised that I would include a review of this book in a few days.  So many months later here it is.  The present book is a revision of Advanced Calculus for Engineers, first published by the same publishing house in 1948.  This earlier edition is now available as a photographic reprint from Martin Fine Books at a modest price.  A second edition was published in 1976, the paperback of which is still readily available - but used hardbacks are available at better prices, especially for the earlier editions.

This is one of the few books that lives up to its name - it really is advanced calculus for applications.  It covers ordinary differential equations and their numerical solution, Laplace Transforms, series solutions, boundary value problems and characteristic-function representations, vector analysis, higher dimensional calculus, partial differential equations, with particular emphasis on those met in physics, and complex variable theory including contour integration and conformal mapping.

This is a very useful book. Don't hesitate to get a copy of any edition!

A First Course in Calculus, 3rd Ed., Serge Lang, Addison-Wesley, Reading, Mass, 1973

This text would eventually go to 5 editions and a reissue of the first edition called Short Calculus.  The edition in the title is the version that I used as a freshman at University College Cardiff in the fall of 1975.  The course was taught by Dr. G.R.H. Greaves.  At the time I thought that Dr. Greaves was in his 40s, but he was actually in his twenties - in fact, he was the last of my teachers still active in the classroom, only retiring a few years ago, and I will be 62 in a few days.

Dr. Greaves was a great teacher and the book was ideal for the course (I took 1A for mathematics majors rather than IB for science majors who didn't plan to pursue a degree in mathematics.)  As a result, I was in a much better course, one that looked forward to real analysis.  It remains a favorite, though since I have had to repair my International Student edition, I more often refer to Short Calculus.  It isn't the book I would recommend to students today - though it is better than most current calculus.  I'd recommend the books by Gilbert Strang or Peter Lax.  Personally, if I were beginning calculus again, I'd like to be in a course that used Peter Lax's book.  I'll write a review at some point.  My all-time favorite remains A course in Pure Mathematics by G.H. Hardy.

The indispensable guide to Undergraduate Research, Anne H. Charity Hudley, Cheryl L. Dickter, and Hannah A. Franz, Teachers College Press, New York, 2017

This is my second review of books about getting involved in student research of the day.  This is the better of the two books.  It is informed by the experiences of the authors and the many researchers with whom they have interacted.  It still isn't the book that I was looking for.  I'm looking for a short book that is equally valuable for students and for their faculty members, particularly new prospective mentors.  I may have commented before that I have found books like the Cambridge Monograph Series of old to be the most useful, and the older ones tended to be between 80 and 120 pages.  At 180 pages, this book is too long.

The book begins with a section on what is research and why do we do it, it then has the following chapters:

1. Finding the tools to become a scholar
2. Getting started with undergraduate research: What, Why, and How.
3. How to fit research in with everything else: Time and Energy Mangement
4. Research with Professors and Mentors
5. Writing and Presenting Research
6. Underrepresented Scholars in the Academy:  Making A Way
7. In Conclusion: Research in Action

This book contains much that is useful - but it is aimed at all disciplines.  If I want a book that focuses on the sciences and mathematics and mentions the latest on funding and the appropriate tools, I think that I will have to write it myself.  I should note that I would have bought a copy of this one - even if I had first borrowed a copy from the library.

Getting In, David G. Oppenheimer and Paris H. Grey, Secret Handshake Press, Gainesville, Fl, 2015

I have long considered writing a book on getting started in student research.  I likely still will.  In the meantime, I'm going to review some books that are already on the market.  This one has the sub-title The Insider's Guide to Finding the Perfect Undergraduate Research Experience.  This one was written from a life sciences perspective.  It has some useful information, but it isn't perfect.  The book has a very annoying feature, it prints the things that it thinks the reader should highlight in bold.  This is also to support skimming.  The book is worth reading - this feature detracts from its usefulness.

The book begins with a section about why students should become involved in research and describes the benefits, both short-term and long-term, that can be derived from such involvement.  It follows this with a chapter on what research is - this likely would have made a better first chapter.  It's tough to get motivated if you don't know what you are doing!  The next chapter is about whether students will like research - some will, some won't.  It does point out that students don't always know at first that they are learning things that will help them.  This gets to one of my major points about this book.  Is it for students, or is it for faculty - the answer is both, but it would have been better to make this clear.  Many new faculty would be helped by a book like this, but it would help readers if the sections for each audience were more clearly identified.  Both audiences need to be familiar with the other's viewpoint, but it would help readers if the information for these roles was better separated.  The authors address the question as to when a student should begin undergraduate research - for me the only acceptable answer is: Now.  What are you waiting for?  The authors describe different scenarios at length, but, in my experience, the earlier the better. 

The book then has a search strategy for finding the ideal position.  This will be helpful for some, but my philosophy on this is that in the beginning topics don't matter - as long as you are going to learn research skills, if you have an offer, and if the environment is welcoming and supportive, then you have found a good place to start.  Over my academic career, I have seen prospective faculty turning down tenure-track offers without realizing that one such offer is all that most people are going to get, and I have seen faculty keep trying to publish in top-tier journals, resulting in insufficient publications for tenure or promotion.  So my advice, if the opportunity is a good opportunity, take it.  You will have the rest of your life to follow your own research directions.  The search and interview process described in this book are also too complicated.  If you are in an environment that is supportive of student research, you can probably get started by talking to your teachers, attending seminars and asking if you can attend some research group meetings.  The CV and interview process described in this book is probably not going to be used for the sorts of experiences that most students are looking for.

Thus, in summary, this book contains much that is useful, but it probably pertains better to students attending major universities rather than the bulk of students who are attending colleges and comprehensive institutions where most students will gain their first research experiences.  It will remain on my bookshelf, but I wouldn't have bought a copy if I had been able to borrow it from the library.  It has some good points, but it has glaring omissions like funding, the availability of NSF REUs and how to progress.