The point of this site is to put together a full roadmap of detailed content -- much like a free e-book -- on mathematics and physics starting at a first-year undergrad level. I'll be describing insights and intuitions (e.g. geometric interpretations) I figure out for myself when reading math and physics, as well as some good problems I come up with, etc. to create what I end up deciding to be the best way to learn these subjects. It can also be considered as a complement to Khan Academy, which is targeted primarily towards K-12 education (I didn't like their linear algebra videos much, either).

This is not the first time I've tried something similar -- in the past, I've created a youtube channel and a mathematics and physics wiki for similar purposes. On hindsight, however, I realise that I tried to speed through my own learning too quickly in order to complete these projects, causing my knowledge of several advanced fields of physics to be superficial and elementary -- like trying to learn gauge theories without really understanding what group bundles were.

The inspiration for the actual style of the content of this site comes partially from Needham's Visual Complex Analysis. It's not so much that there aren't other books like it, including for other topics, but rather that resources tend to present the topic in a single way -- either purely intuitively, or purely rigorously, or whatever, and either a pure undergraduate-level textbook, or a pure graduate-level textbook, etc. I hope the resources on this site will be able to cover all the different possible requirements that readers might have -- covering the insightful, rigorous, historical, intuitive, conceptual and computational aspects of mathematics and physics -- and will make full use of the choice and power that the Internet gives to resources such as these in order to achieve this purpose.

Credits: All content created by Abhimanyu Pallavi Sudhir. Software attribution:
  • Runs on free Blogger software offered by Alphabet, Inc. (aka Google)
  • Widgets used: Blogger Random customisable feed widget

How to use the site

I wouldn't actually recommend reading "in order" on the the site -- i.e. don't go semester-by-semester, year-by-year -- the order there is just there to get you an initial sense of ordering of the topics you'll be learning on The Winding Number. There is no hard-and-fast order to reading the topics, but I would make a few prescriptions:
  1. Try to read corresponding math and physics topics together, even if you only care about the math. For example, learn linear algebra along with special relativity (then while thinking about more advanced, abstract linear algebra, do quantum mechanics), calculus of variations alongside Lagrangian mechanics and other advanced classical physics, differential geometry alongside general relativity, etc. All too often, you'll find that mathematical topics correspond very neatly with physical topics, perhaps for historical reasons as to why mathematics is developed or why a certain area of mathematics becomes interesting and popular to learn. Doing these side-by-side makes you able to transfer insights between the corresponding topics and gets you an appreciation for why specific ideas are important, etc.
  2. Don't try to read the site -- or any resource for that matter -- line-by-line, word-by-word etc. That's not how you learn things, and you become reliant on other people thinking for you. My preferred method of learning things is to skim through texts, to get an idea of what the topic is about, motivate (genuinely) each idea you find and work out the actual meat for yourself (and only if you're really, really stuck -- like even after a month of trying -- should you refer to other people's insights, such as mine at The Winding Number -- it's okay to get little-known proofs and such from me). This means you have to put your shoes in the face of the person -- sorry, put yourself in the shoes of the person actually discovering the idea, which trains you to do original research yourself, or for any job that requires original thinking. You could also discard the textbook entirely and derive everything from scratch, but mathematics is 4500 years old, so this would require 4500 years, so in lieu of that the "skim-and-work-out" method allows you to stand on the shoulders of giants a bit, but you need to climb there yourself.
  3. Proof-writing is overrated, it should come to you naturally if you understand what you're proving, you just "formalise your intuition", because your intuition should be logical. Proof writing is essentially a language designed so that you don't need to bother thinking about how to convey your insights, instead expecting the reader to figure them out for himself from reading your proof. It's like an operational summary of an Heinlein novel, or a technical drawing of a picturesque sunset. I'd start with formalising proofs of some things in calculus -- stuff like convergence tests, trivial theorems (mean value etc), some basic real analysis (Rudin is a good resource to follow), etc. Then go to the very basics, and think about how you could formally define stuff like multiplication and exponentiation. Once this is done, you should be able to efficiently write any proof you know.
  4. Several friends and online followers of mine have often asked me questions to the style of "What book should I read to understand such-and-such subject?" -- this kind of questions are as bad for your brain as drug abuse or alcoholism. In reality, what book you use matters much less than your own style of thinking. Focus on finding the right insights in physics and mathematics through your own original creativity, on thinking operationally in programming and computer science, thinking precisely, fundamentally and abstractly in philosophy and economics. Questions regarding resources should be framed as "What resource should I follow to get an idea of the structure of a subject?" The content should come from your own ability to think.

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    Specific textbook recommendations can be found on the related module pages.

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