I am interested in visualizations/simulations of physical systems as a way to learn advanced math. Is there any books or resources that take that approach?
https://ocw.mit.edu/courses/res-18-008-calculus-revisited-co...
When looking into simulations of physical systems, you'll run into partial differential equations, but be careful about learning resources that don't put numerical methods front and center. The article on Numerical Weather Prediction in the post has a good description:
> "Analytical solution of the equations is impossible, so approximate methods must be employed. We consider methods of discretizing the spatial domain to reduce the PDEs to an algebraic system and of advancing the solution in time."
Given Python's popularity in scientific computing, a lot of the available materials on the topic are in that language, using libraries like numpy and scipy a lot. I've been playing around with custom ChatGPT here - you can construct a workflow that takes a description of a common equation, generates the LaTex expression for it, translates that to a sympy expression, and then from that generate the numerical method code using numpy and then the code to plot the behavior over a given range in matplotlib. Bonkers, we're living in the future.
I think most differential equations courses are too focused on symbolic solving techniques. To me, understanding a (physical) system is mainly understanding the differential equation (system) itself, not it's solution. MathTheBeautiful really excels at this approach.
Solutions are of course important as well, but i think that's what computers are for.
[0] https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCV...
Uses Haskell, and teaches it so knowing Haskell is an unnecessary prerequisite, though familiarity with programming in general would be very beneficial, and focuses on 2D and 3D visualisation.
I learned Penn because it's "real" (mainstream) math & lets you read in the same language; I learned GA because it's designed to proselytize to other mathematicians & deliberately presents serous ideas in very approachable ways.