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I'd like to learn mathematics at a (semi-?)professional level.

I'm studying Undergrad Medicine in South Asia. I'm looking to build up on my understanding of mathematics from scratch. This is to help me better comprehend the Physics and Chemistry that makes up much of Medicine (besides, I've taken a liking to pure Mathematics anyways).

To this end, I wish to follow (more or less) the curriculum prescribed for Undergrad Math courses in my spare time. To elaborate: I feel the best way for me to grow as a student of mathematics would be to adopt a typical Undergrad Math course's curriculum, and work on it in my spare time.

However, I have absolutely no idea how an actual Undergrad math course is structured to begin with.

I did try to look up university brochures/handbooks for the course, but as someone with no exposure to mathematics-education outside of high school, I'm not able to make much sense of it.

enter image description here (This snippet was taken from Oxford's Undergrad Math course handbook. It wasn't particularly enlightening to me).

So if it isn't too much to ask, I'd greatly appreciate it if the Community at Math.SE can offer me guidance in this regard. So, I suppose my question ultimately boils down to:

What's the pedagogic hierarchy of various sub-disciplines in Undergraduate Mathematics?

(Should I start with Algebra? If so, what kind of Algebra? Should I follow up with Geometry? Or should I read them side-by-side?) Hopefully, you get what I'm on about.


Addendum:

1) I'm not requesting book recommendations here (I'm aware there's a separate, dedicated post for that on Math.SE). However, I'd still appreciate recommendations, should you have any :-)

2) I'm aware of "capsule"-books like Mathematical Methods for Physics and Engineering, however, I wish to acquire a far more rigorous understanding of the subject. This, I feel, can only be accomplished by actually following the curriculum prescribed for a generic Undergrad Math course.

3) I'm aware of other related (but different) questions on this site. For example- (1), (2)

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    $\begingroup$ The pedagogic hierarchy is extremely country-dependent. In the same way, what you “should” (there are probably several good choices) start with depends on what you already know (what a bijection is? What complex numbers are? how comfortable are you with mathematical proofs? and so on). $\endgroup$ – Mindlack Sep 2 at 12:05
  • $\begingroup$ @Mindlack Nods I suppose I'll have to wait for multiple answers/suggestions and then pick the one that suits me. $\endgroup$ – paracetamol Sep 2 at 12:32
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    $\begingroup$ @paracetamol In determining where you should start, it would greatly help if you could say what your current level is. You say you’vd only done math in highschool, but what type of math is done in highschool and how much of it do you remember? $\endgroup$ – Ovi Sep 2 at 13:03
  • $\begingroup$ @Ovi Point. I'll try to edit the post accordingly! :-) $\endgroup$ – paracetamol Sep 2 at 15:13
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Linear Algebra I, Analysis I, Probability and Geometry are foundation level courses and can be studied in any order - or indeed, in parallel. Other foundation level topics could be Groups, Topology, Logic and Number Theory. Introductory Calculus will be applying the results of Analysis I (or you can think of Analysis I as being the theory behind Introductory Calculus).

More advanced topics then build on these foundations - so Statistics builds on Probability; Complex Analysis applies the techniques of Analysis to functions of a complex variable; Algebraic Number Theory applies Groups and Modern Algebra to Number Theory; Analytic Number Theory applies Complex Analysis to Number Theory etc.

So there is no single linear order, but more advanced topics will assume an understanding of other subjects as a pre-requisite.

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Before studying anything else more advanced than Calculus (at least American Calculus, not sure about other countries) , you must 100% make sure you have a good understanding of basic mathematical logic and basic set theory: "the symbols $\forall$ (for all), $\exists$ (there exists), the logical connectives, and etc. One good place to learn all of this is in a Discrete Mathematics book. The book I used, "Discrete Mathematics" by Kenneth Ross, covered logic in 1 chapter, sets in another, and functions (which I would also recommend) in a third.

After this, you can do pretty much anything you want. I would perhaps say that it's better to take Linear Algebra before Abstract Algebra if you can help it.

As far as books, I would say that (if possible) to look at multiple books and see how the material is presented in different sources. One good source I like to read is Wikipedia. Often times in a book you can get bogged down in the details, but Wikipedia provides a bigger picture, often relates the topic to other topics, and provides some history. For me, this can help a lot.

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  • $\begingroup$ +1 Clearly I've got to brush up on my basics. Thanks for the book recommendation! :-) $\endgroup$ – paracetamol Sep 3 at 3:42
  • $\begingroup$ "...Wikipedia provides a bigger picture, often relates the topic to other topics, and provides some history." I'd give you a +10 for this if I could ;-) $\endgroup$ – paracetamol Sep 3 at 3:44
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    $\begingroup$ @paracetamol Sure, hope this helps! :) Like I said I’ve only used one discrete math book, so I don’t know if its the best, but it did the job for me. $\endgroup$ – Ovi Sep 3 at 12:33

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