Does any of you know a book with mathematical problems from various fields? I'd like to have a single book that contains problems from calculus, topology, probability etc.
I assume you are looking for student-level problems, as opposed to open research problems (the latter are probably given separate books according to subject).
Berkeley Problems in Mathematics. Pdf version here. Roughly, it's a book of problems, sorted by subject, all taken from past UC Berkeley graduate qualifying exams - so at least in principle it's a good test of your knowledge of undergraduate topics. The topics are as follows:
- Real Analysis,
- Multivariable Calculus,
- Differential Equations,
- Metric Spaces,
- Complex Analysis,
- Linear Algebra.
You will note that it's quite analysis heavy; this is just a quirk of the way the book has been put together: if you want more algebra, there's probably another book for you.
There is also the Stanford Mathematics Problem Book, but I don't know much about that. It might be worth checking out, though.
Perhaps it could be useful to also look for reference books that cover a wide number of fields, so I would like to add the book Mathematics: Its Content, Methods and Meaning by Kolmogorov, Aleksandrov and Lavrentev, which covers linear algebra, functional analysis, group theory, ordinary differential equations, the theory of functions of real and complex variable, and I think that filling the gaps would be a good excercise.
As for a book on exercises, I'd like to add a book on problems on probability, 'One Thousand Exercises in Probability' by Grimmett.