Learning mathematics for physicists from scratch

Question

i am a freshman physics student and naturally my curriculum includes math-classes. The thing is, that -at least for the time being- teachers cover only the surface of topics so as to have only a practical knowledge of math. However i understand the philosophy behind that strategy, i find mathematics too beautiful to confront them as only a box of tools for my physics. More over i believe that the deeper in understand mathematics the better theoretical physicist i will become.

So, to be more explicit this semester i studied linear algebra. Since my book covered as i said only the basics i naturally researched the internet. However, the deeper i got the more unknown topics and definitions poped up like tensors, permutation group,metric spaces etc some of them i later got an idea of .

So what i would really like to ask is, if there is a right sequence of topics to start learning deep mathematics that will eventually help me with physics as well.

For example: I know that calculus is like the bible to a physicist...so i would like to study for example the principia at some point because i would like to know the very basics of calculus as well.

Another Example:I really liked linear algebra and i know that a physicists makes use of it a lot as well. Should i have studied something else before linear algebra to make more sense to me? for example group theory etc?

Hope i am making some sense. Thanks in advance.

Answer 1

In response to your question about calculus for physicists, I would suggest you study the book by Keisler entitled Elementary Calculus and freely available online here.

The book uses an intuitive approach exploiting infinitesimals and is particularly well-suited to a future physicist who is less concerned with epsilon-delta techniques that many math majors find confusing but which are considered a sine-qua-non for a well-rounded mathematics education. Since you seem to plan to study physics this may be ideally suited for you.

For more advanced applications of Robinson's framework to physics, you can see the book

Albeverio, Sergio; Høegh-Krohn, Raphael; Fenstad, Jens Erik; Lindstrøm, Tom. Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986. xii+514 pp.

Answer 2

As one who started in physics major and ended up being in math, my experience is that math is much more difficult than physics at undergraduate level. That said, basic calculus and linear algebra education should be the same in either division. At this level, any textbook would pretty much serve well enough. As long as you get decent grades, you in a good shape. It may be worth mentioning that mathematics is not a subject learned via the old classics, unless you are interested in its history. It is more important to adopt the modern viewpoints.

The fundamentals of undergraduate math are analysis and algebra. Both will study abstract objects axiomatically. For a taste of these, my suggestion would be to read an advanced book on linear algebra, like the one by Hoffman and Kunze. In particular, you should study vector spaces very carefully. It is profound in nature, crucial for both math and physics, but relatively easy to learn.

Answer 3

Depends what are your aims. Indeed I think that the best thing you could study as freshman in physics is basic group theory.

In Italy we do that in first year of Mathematics taking a two semester course called "Algebra", while often is not in a first year of Physics. Anyway modern Physics relies deeply on Algebra and so knowing a deeper notion of groups, field and some algebraic tecniques will turn out to be even more useful in nowdays Physics then plain calculus.