Physicist Amok

A friend who took his degree in mechanical engineering recently asked me for a plan of study to solidify his mathematical knowledge. He is already competent at multiplying matrices, integrating, and churning out solutions to the common families of linear differential equations. At the same time, he has no need of Bourbaki. Here is roughly the curriculum I outlined.

Start from calculus again. The introductory calculus book has one purpose: to get the student ready to work through baby Rudin. My recommendation is Spivak’s Calculus. Michael Spivak is one of the great mathematics writers of our day, and his book is designed to fill exactly this niche.

Once you have finished Spivak, it’s time to tackle Rudin’s Principles of Mathematical Analysis (a.k.a. baby Rudin for its small size). Rudin’s book is not fun, but you don’t need to remember everything in it after the fact. You do have to get comfortable with set theory, topology of the real line, sequences and series, and how to use these tools to construct proofs in analysis. Later this book is an invaluable reference. You might also put off the study of functions of several variables and integration of differential forms to Spivak’s Calculus on Manifolds below.

Linear algebra is next. Axler’s Linear Algebra Done Right is probably the place to go for this. The important thing is to learn the structure of linear maps. Matrices are just a computational tool — albeit a vital one — for dealing with such maps.

With analysis of one real variable and some linear algebra behind us, multivariable calculus is next. Spivak’s Calculus on Manifolds is probably the nicest book on this that I know. It’s also very short.

Complex analysis is the other important field at this stage, and it can be done before, in parallel with, or after multivariable calculus. Tristan Needham’s Visual Complex Analysis is an extraordinary exposition. He has managed to draw pictures of expansive parts of the field. The detailed analytical machinery tends to get short-changed, however, so I suggest combining it with a very terse complex analysis book of a more traditional character. You might look at the recommendations of Stephen Greenfield, though this is for graduate complex analysis.

Now we come to ordinary differential equations. I find V.I. Arnol’d’s Ordinary Differential Equations charming. You’ll need a couple things for this. First, everyone needs some group theory. Humphrey’s A Course in Group Theory is a fast paced exposition with a single goal: the classification of groups of order less than 32. It’s very clear. Then you’ll need differential geometry, and you could hardly do better than the first one or two volumes of Spivak’s five volume Comprehensive Introduction to Differential Geometry. He’ll also give you the material on Lie groups.

Unfortunately, from here I haven’t really explored the books, and what I know I learned directly from people.  But by this point, the student’s ready to be out on his own.