I thought I was just drawing pretty lines, but my high school doodles turned out to be Linear Algebra over a finite field. When I was in high school and college, I would always try to read ahead in my classes so I’d be prepared when the topics came up…

We’re going to introduce fields in preparation for finite fields and show how they answer questions that went unanswered for millennia — In the last article, we got to the point where we could define polynomials using rings and we could have solutions to those polynomials with division rings. These solutions, however, were difficult to work with. Previous Article Next Article All Articles To solve this problem, we require that × be…

Lagrangian Mechanics makes finding the equations of motion almost trivial and will serve as the basis for modern Physics. As I’ve said before, there are many tasks that can be quite difficult in Newtonian Mechanics. These tasks include finding conserved quantities, dealing with constraint forces, converting the equations of motion…

Groups may allow for solvable equations, but we need more structure to even talk about polynomials. We call these structures rings. In the previous article, we established groups as the simplest algebraic structures with solvable equations and we showed that the five color group was isomorphic to the finite Abelian…

Differential Geometry is the study of Calculus on smooth manifolds, and it allows you to make statements that are true in any coordinate system. At this point on the Road to Quantum Mechanics, we know how to set up and solve some important partial differential equations in different coordinate systems…

Communication is a skill that everyone should take seriously, especially scientists and science communicators. — Recently, Rebecca Lea Morris published an article called Ingrid Daubechies and the Importance of Mathematical Communication. She tells the story of how Ingrid Daubechies published a paper on wavelets that led to them becoming an industry-standard in several industries. Daubechies cites her attempt to make it easy to read and…

We’re at the end of the PDE arc. Let’s use the techniques we’ve learned to derive and solve the Wave Equation. — We’ve talked about the Wave Equation before in passing and in a few exercises, but we’ve never done anything with it. In this article, we’re going to fix that. We’re going to use every tool we’ve developed in the PDE arc to derive, solve, and analyze the wave equation. Previous…

You’ll gain a lot more than you’ll lose, even without considering the negative effects of climate change. Few, if any, specific plans have been drafted about what a sustainable future look like. At best, we have vague ideas that we should switch over to renewables and recycle and maybe take…