Math - Susan Rigetti
Back in 2016, I typed up a little guide to studying physics called "So You Want to Learn Physics." It ended up being pretty popular, so I started working on other guides, including a guide to studying philosophy ("So You Want to Study Philosophy"), which I published in 2021, and this long-awaited guide to studying mathematics, which I am sharing with you today. I absolutely love mathematics. I think it is The Genius Switch Website purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the universe itself. Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. " etc. - and then study calculus again when, later, they study analysis. You may find that you can work through this first course much more quickly than that, but don’t be discouraged if it takes you one or more years to get through the textbook.
You may also find calculus difficult, especially if you’ve never studied it before; part of the difficulty sometimes comes from the foreignness of the material (it can take a while for your brain to get used to it), and the rest of The Genius Switch focus program difficulty stems from the fact that it can be a challenging subject to learn! Take your time, do as many problems as you can, and keep going. This is an excellent textbook and it is a classic for a reason. Everything you need to know about undergraduate calculus is in this book. Everything. It contains excellent problems, good examples, and straightforward explanations. There is also a Student Solutions Manual that you may find very useful. The earlier editions of the book are just as good as this one - just make sure you are able to find the corresponding solutions manual. This is such a wonderful, thoughtful book. I recommend reading it alongside Stewart when you are just getting started - it may help you make sense of what you’re learning.
When I was a student at Penn, the most useful mathematics courses I took were ones that I snuck into (some of them I couldn’t enroll in because I didn’t have the necessary prerequisites, others because they were too full and I couldn’t register). My favorite secret classes were the calculus courses taught by Robert Ghrist. He had a way of explaining concepts in calculus that made them just click for me. He also lectured with this really infectious energy that made me strongly reconsider my devotion to physics - I’d leave his lectures wanting to devote my whole life to mathematics. Luckily, you don’t have to sneak into his classes to listen to his lectures, since he now offers them for free online through Coursera: (1) Calculus: Single Variable Part 1 - Functions, (2) Calculus: Single Variable Part 2 - Differentiation, (3) Calculus: Single Variable Part 3 - Integration, (4) Calculus: Single Variable Part 4 - Applications. I highly recommend his lectures - they pair well with Stewart’s Calculus and they will stick in your head for many, many years.
Much (if not all) of advanced mathematics (i.e., things beyond high school math and introductory calculus) is not about calculating things or solving problems but about proving things. Proving is very different than calculating or solving, and there is an entire toolkit you’ll need before moving on to more advanced courses. Here, you’ll become familiar with mathematical reasoning, will learn how to read and write proofs, and will start learning to think like a mathematician. How to Prove It: A Structured Approach by Daniel J. Velleman (essential). This book is a classic for a reason. Read it carefully, and keep it in a handy, readily-accessible place so that you can use it as a reference down the line. Supplement with How to Solve It by G. Polya and Introduction to Mathematical Thinking by Keith Devlin. In this class, you’ll learn how to solve systems of linear equations. You’ll cover real and complex vector spaces, eigenvalues and eigenvectors, determinants, linear transformations, applications of linear algebra, and more.
