11–20 of 231 posts
It's like when you have Calculus and over generalize when entering Analysis or Differential Geometry -- but now I have more structural patterns to extrapolate from :/
You will not improve your math watching youtube videos and reading books. You need to produce stuff that pass the "sniff test" to your colleagues.
Although Math 55 is tough, if you are self-paced and have a bit of mathematical maturity I think it is doable. It's also an excellent pure math bootcamp that gives you a solid foundation to branch into any other pure or advanced math topic.
I have gone through Halmos & Rudin myself, and it is a great experience. However, if your end goals are more geared towards pure CS, an alternative route might be much more appropriate. Very interesting and promising parts of CS, such as formal methods, and the foundations of mathematics themselves depend on abstract algebra and logic: https://ncatlab.org/nlab/show/computational+trinitarianism
A minor problem is that beginner literature is not so polished as it is relatively young. But there are some excellent textbooks nonetheless. Some below. Other suggestions welcome:
* http://www.cs.man.ac.uk/~pt/Practical_Foundations/
* https://www.mta.ca/~rrosebru/setsformath/
* https://github.com/ademinn/ttfv/blob/master/2006.%20Sorensen...
* http://www21.in.tum.de/~nipkow/Concrete-Semantics/
1 Every day on this site https://www.khanacademy.org
2 Use this for graphing desmos.com/calculator
3 Use this to supplement (ie free text) myopenmath.com
I honestly don't think there is better mathematical content than his being made.
The hard but maximally useful thing to do, in my opinion, is to regularly meet with at least two other people and a blackboard and beat your heads against it together for about 3 hours at a stretch. Do this at least weekly--and preferably more often. Self study in between meetings is obligatory. One of those people should be at about your level. The other should be farther along. All three of you should trust each ot…
If we are talking about content you can find plenty of advice here or elsewhere. But do yourself a favor and pay a tutor and/or find a study group. As in writing, dancing, etc. you cannot evaluate your own work good enough. You will not improve your math watching youtube videos and reading books. You need to produce stuff that pass the "sniff test" to your colleagues.
That depends on exactly what you mean by "become good at math". If you're talking about "becoming a mathematician" and doing original research in pure math, then you're probably right. But if one means "learning existing math well enough to apply it to a problem", I would argue that one can learn this stuff just using books, videos, etc. At least up through a certain level.
That said, I do encourage the idea of finding peers to work with. I used to coordinate a "math study night" at the local hackerspace for that exact purpose. It fell off because I got busy and couldn't keep committing to it, but generally speaking, it is a good idea to have other people to work with. I may well try to find a math major from UNC to hire as a tutor at some point as I keep working on this stuff.
And if one can't find somebody to work with in person though, and they need, say, a proof evaluated, there is the option of using math.stackexchange.com or the like.
The hard but maximally useful thing to do, in my opinion, is to regularly meet with at least two other people and a blackboard and beat your heads against it together for about 3 hours at a stretch. Do this at least weekly--and preferably more often. Self study in between meetings is obligatory. One of those people should be at about your level. The other should be farther along. All three of you should trust each ot…
Just to add, it's also worth finding someone who might be a bit below you in terms of maths ability. If you can explain the concepts to that person, then you know the concepts.
The University of Oxford has all of their notes and exercises available online https://courses.maths.ox.ac.uk/overview/undergraduate If you get stuck you could try asking on https://math.stackexchange.com/ Although, if you can, finding someone else to work with, and someone else who already knows some math to ask occasional questions would probably help you a lot.
Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically compute quantitative information about it (its value at 3 is blah, its integral over this interval is blah, its maximum value is blah).
Linear algebra: vector spaces and linear operators, and their representation as vectors and matrices. Functions as forming infinite dimensional spaces, and Banach and inner product spaces.
Differential equations and dynamical systems: extending what we did for calculus to differential equations. Phase space, orbits, Fourier and Laplace transforms, sets of linear differential equations, numerical integration, some partial differential equations. You do not need all the little tricks for special kinds of equations that you will find in, say, Boyce and dePrima. They're not helpful.
Sets, groups, rings, and lattices. Mathematics today is written in terms of set theory. You need to understand the basics of manipulating sets and functions between them. Then you should know something about groups, rings, and lattices, which are the most ubiquitously useful algebraic structures besides vector spaces.
After that, where you go is going to vary enormously. Based on what you're aiming to do.