A little background for my question: I have been studying Math with a problem/solving strategy until very recent (mostly taking a bunch of interesting problems from contests like IMO and traying to solve them, and reading Engels' problem solving books), and so I feel comfortable with the basics methods for proving, and also with the basics of Number Theory, Geometry, and Discrete Mathematics. So far I have studied a little of Calculus (just finished Single Variable),and Probability Theory. Recently, I began to study Discrete Mathematics with a main focus on Combinatorics and Graph Theory, and I plan to start both Linear Algebra and Differential Equations as soon as I finish Multivariable Calculus.
I love Mathematics, specially what I have seen from Applied Math (although my background is very reduced yet), and I've been trying to build like a roadmap to self-study the subjects that I need (using the MIT Math major requirements). I will start college in about a year, but since I have the time I want to prepare and to learn as much as possible as I really enjoy Math, however I've been struggling with some of the technicalities of this idea of self-studying the major:
I'm not sure about which subjects I can start to study in parallel as for example both Linear Algebra and Probability Theory require Multivariable Calculus. I don't know if it suffices to study them at the same time of Multivariable Calculus, or if I should first study Calculus and then start the other subjects. And the same with other classes, I don't know what subjects I can start and until where to reach at a first encounter, and also where to go once finished. For example I don't know at which point I should start Real Analysis. Hence a roadmap for the Math major with some tips of what classes to take first and which of them I could take at the same time would be very helpful.
Any recommendation about the textbooks and materials (articles, books, courses, lecture notes, $\ldots$) to use for each subject is also very appreciated (I know that this particular point has been debated before in this forum but any new opinion is very welcome). I've been using MIT Opencourseware website so far and here is a list of the textbooks that I've been using for self-studying as well:
- Calculus: A Friendly Introduction to Analysis (Kosmala)
- Probability Theory: Introduction to Probability (Bertsekas and Tsitsiklis)
- Discrete Mathematics: A Walk Through Combinatorics (Miklos Bona)
- Abstract Algebra: Abstract Algebra (Foote & Dommit)
- Linear Algebra (soon to begin): Linear Algebra (Friedberg, Insel, and Spence)
- Algorithms and Data Structures: Introduction to Algorithms and Algorithm Design
Thank you for taking the time to read this!!! Any comment/critic/advice is welcome =)
My understanding is that you've learned single-variable calculus from Kosmala's analysis book and you'd like to know where to go once you've finished the rest of his book.
I'm not familiar enough with Kosmala's book to know whether it has enough practice with basic calculus problems. If you think it doesn't, you could use Maron's Problems in Calculus of One Variable alongside it.
After that, I'd say that the schedules of the undergraduate math program at Cambridge are likely to be a more transparent guide than the MIT undergraduate program as to what to read in what order, particularly as you say you're interested in applied math. Here are the most recent schedules: https://www.maths.cam.ac.uk/undergrad/files/schedules.pdf
And lecture notes: https://www.maths.cam.ac.uk/undergrad/studentreps/tripos-specific-resources
I believe you'll find that the physics books suggested there and in the lecture notes of some of the faculty (see David Tong's webpage in particular) are better suited to applied mathematicians (as opposed to physicists) in comparison with undergraduate physics courses taught in the United States.
I would sugeest that your next major steps should be to learn algebra (I really like Artin's book - it combines abstract and linear algebra nicely), and analysis at a higher level than in Kosmala's book (for example, Apostol's analysis text is great, and the second volume of Zorich should also be possible).
Before then, I'd try to learn enough about vectors and complex numbers from a geometric and algebraic standpoint to be comfortable when they come up in algebra or analysis, though you might find that Chapter 1 of Apostol's text is enough for you on these topics.