Get Thomas 13th or 14th edition and solve all the odd problems. That takes care of calculus.
Ask HN: How to self-study mathematics from the undergrad through graduate level?
131–140 of 231 posts
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#132I'd very strongly recommend https://en.wikipedia.org/wiki/How_to_Solve_It to familiarize yourself with some fundamental processes and https://www.amazon.co.uk/Playing-Infinity-Mathematical-Explo... before you lose yourself in a highly symbolic and abstract world, appreciate the beauty of it first, it'll make the long journey much easier.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#133I know this is not what you're asking for, but I'd recommend stopping at the end of the sophomore level classes in the open courseware material.
I finished an undergrad degree in math from a school that had a large PHD program, and once I got to the junior level it was a lot of proving obscure facts about complicated objects (say, Lie Algebras) that have zero applications outside of very speculative theoretical physics. Proving theorems was fun sometimes, so if you're interested in math as entertainment, it's worth trying the more advanced material.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#134There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices. 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 comput…
Any suggested reading on rings and lattices? I took a lot of math classes in undergrad but was never exposed to these concepts.
Really enjoyed Algebra: Category 0. Introduces some basic Category Theory and uses it to develop the standard practice of undergrad algebra. The exercises are medium difficulty (nothing ridiculous like a Knuth or Spivak problem, but nothing breezy either) and there are enough examples in the text of proofs with enough detail to get you working on the exercise problems.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#135As an alternative I would suggest a top-down approach. Start with the theorems/results you truly wish to understand and work backwards. There was a great quote from an interview of Peter Scholze (one of last year's Fields' Medallists), which has really changed how I view learning: At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which…
The issue with this approach is that if you learn just "the fun parts" you might be left with some huge gaps in your knowledge, all that in-between stuff - especially if you're not a genius like Scholze. Standard approach is perhaps less motivating and you learn a lot of stuff that frankly you'll never need and you'll probably forget most of it, but it ensures that you've at least heard about all the major ideas. One…
I think fun is an extremely important part of keeping up the pace of learning. I would rank that much higher over "methodically combing through everything", which could cause you to quit from boredom long before you reach your goal.
I'd hypothesize that the problem you describe is due to insufficient curiosity about the way things are already done. Especially with Google, Wikipedia, and well-populated internet forums, it'd be hard not to find "foundational solutions" after a little research.
I had a music teacher who was obsessive over minutiae, and fussing about the exact position of each finger on a piano key was much less fun than loosely noodling around with chords and making things that sounded closer and closer to real music. I made little progress with the lessons and quit, but since I started noodling on my own I've been getting much farther.
I propose that "intelligent" is a synonym for "gets enough pleasure from learning to do a lot of it", so I would overwhelmingly optimize for that.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#136There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices. 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 comput…
Most engineers I know have not learned set theory or groups/rings/lattices. They still seem to be doing pretty well.
Another thing most engineering courses skimp upon is variational methods/calculus of variations, and that has effect in how they do classical mechanics. Physics majors and engineers do classical mechanics in very different manner.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#137I discovered this amazing YouTube channel recently that I wish I had access to back in college. It explains some very difficult math concepts in a very simple and easy to understand way: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#138I think it is pretty much impossible to learn mathematics without a teacher beyond a certain point, multiple teachers are ideal. The reason for that is in order to do mathematics you need to solve problems and develop intuition. It will be very hard for you to solve problems (write proofs) without someone that evaluates the quality of your solutions, and it is very easy to deceive yourself that your reasoning is without gaps without experience. The same is true for intuition, without a master at whatever field you want to study it will be hard to work through most books (Hartshornes, Algebraic Geometry comes to mind. His seminars and lectures contained plenty of simple examples. The book itself is famously an uneasy compromise, which no one was really happy with but also with few better alternatives)
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#139Earlier quoted context omitted.
Like Einstein's Olympia Academy! https://en.wikipedia.org/wiki/Olympia_Academy
Can we start something like this online for Hacker News community who are interested in Mathematics? I have created a group here if someone is interested to join: https://groups.google.com/d/forum/projectfermat (If you think that there is a better place to have a forum like this that anyone can easily view or participate in, please let us know. I mean we could also create an IRC channel, Slack workspace, etc. but the…
I'd be very interested in something like this.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#140I discovered this amazing YouTube channel recently that I wish I had access to back in college. It explains some very difficult math concepts in a very simple and easy to understand way: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
It's amazing how seeing something visually or having someone else solve something can aid in your understanding of a topic. I remember having the most difficult time understanding mergesort from my book's pseudocode and my professor's lecture. But seeing mergesort in action on youtube revealed what it truly was and made what seemed impossibly complex, so simple and elegant.