The minute I stepped into college I started self learning a lot of math -- Below are some of things I studied. Some methods I tried and some other excellent resources I think people should look into. See what works for you or lemme know what worked for you and you want to add in :))
As Terence Tao says, everyone has there own unique strong point of translating mathematical information into neural signals, the approaches might vary. But one thing is certain - Math is learnt actively (write and solve) and iteratively -- it never the particular problem but the abstract idea that, is signalled through the problem is more important.
1 Real Analysis
Yep u heard me right :P. The first thing I did was RA. Here is an excellent course by MIT OCW. Although this is introductory, it gave me very strong foundations on proof writing. I did nothing much saw the videos and did the assignment as if I was taking the course myself. Understanding theorems can get challenging at times but I think RA serves as :
- A good intro to pure math
- Theorems of sequences and series are used in applied continuous math -- a lot in Electrical Engg!
2 Linear Algebra
This is something that I m still studying and will keep studying :)) The more u delve into the more u find. 3B1B Essense of linal Playlist is a MUST !!
There are two sides on Linear Algebra:
(a) Abstract Linear Algebra -- the study of a special type of objects in math known as vector spaces with more emphasis on algebraic properties of vector spaces in general. Artin is an excellent book for this! But this is better studied after studying Abstract Algebra (b) Applied Linal - The Gilbert Strang Linal - yupp u read it right -- infact the best way to study wud be to pick up Gilbert's book and go thru his course! I studied this from random sources and was never structured -- more like I picked bits on the way
3 Discrete Math
Unlike the Standard Rosen path I learnt DM in tid bits -- cardinality and number systems from Real Analysis. Did not study a lot of combinatorics and for the remaining Prof S Akshay has some cool slides
4 Abstract Algebra
One of the most beautiful things I have ever seen in my life -- Everyone should give a shot to learning how algebraic sturctures with the most primitive properties give rise to the strongest cryptographic protocols. Nevertheless, the subject is just cute in its own right!
Hard Way: I N Herstein : Caution for those of you who've solved Irodov's physics problems. This is the algebra version!! The Simpler but still elegant way : Contemporary Abstract Algebra by Joseph Gallian
I enjoyed it so much that I took a course even after learning it :P
5 Probability Theory
Not talking about poker puzzles or market strategies (sorry quant folks) -- the theory of setting up probabilities as a notion of the chances of something happening -- modelling and doing the math from scratch
Berstekas and Tsikilis is a very beautiful and elegant book -- doing its problems not only widened my understanding of probab but imporved my math! Worth spending time because they force you to apply other areas of math :)
6 Graph Theory
One of the hardest courses I had at college -- self learned through Bondy and Murthy though I dont think it is too hard for an introductory course and focuses less on the abstract ideas. Very proof oriented -- a lot of ideas are elegant but I did not find it particularly beautiful - not a combinatorics person :|. For those using BM, this solution manual I wrote might help. Doughlas B West is also good imo
7 Tensor Analysis
Something non trivial -- not many people talk about this -- I was just fascinated how would matrices would look in higher dimensions. Studied from Chapter 10 of Mathematical Methods in Physical Sciences by Mary Boas. Dealt with standard tensors in Physics + Coordinate transformations in Cartesian and non Cartesian Systems.
There is a bunch of other stuff I tried learning but failed -- topology (Munkres v good, 3B1B must), functinonal Analysis (tried 3 times died after 2 lectures), Statistics (kept finding the right book), Convex Optimisation (Tried Numerical Optimisation by Nocedal and Wright -- helped me thru matrix calculus tho),
Cool channels for learning:
Cool channels for intuition:
An Infinitely Large Napkin book by Evan Chan contains a lot of pieces connected together + provides intuitive insights into abstract concepts