Mathematics
Background
Certain higher-level mathematical concepts didn’t come easily to me in elementary school. Once I had a solid understanding of basic concepts like algebra and geometry, others including linear algebra, trigonometry and calculus came much more easily. Math really started to interest me once I began to realize the plethora of real applications that all types of math are used for and how much of daily life is dependent on mathematical functions. Its applications in physics, both real and theoretical, chemistry, finance, data analysis and media editing among other things all individually contribute to the world that we know, and that idea fascinates me.
I started to take math education quite seriously in high school calculus when the concepts that I had learned previously came together and made more sense. Derivatives and integrals play such a massive role in physics, which I was growing to appreciate at the time, eventually leading me down the path of engineering. After passing my Calculus AP test and earning college credit for my high school completion of Calculus 1, I moved to “Math 2” in college which involved more advanced calculus concepts like swirl and curl, second and third derivation, and double-integration. Following that, during my second semester, I learned to solve linear systems and about concepts such as LU factorization, diagonalization and eigenvalues.
In my sophomore year I took both differential equations and vector calculus, both calculus-based and quite difficult, though again, interesting and extremely useful. I then finished off the requirements for obtaining a math minor by completing a numerical analysis class which was probably the most difficult and content-packed class that I took at the university. We covered a broad range of computation-rooted topics focused around IEEE standards including root estimation methods, error approximation methods and rates of convergence of those methods.
Although I had earned my math minor at that point, I continued with math-related classes, finishing off my senior year with a control systems class. The class involved using previously learned differential equation solving methods and stacking more difficult math on top to solve for control system variables and stable variable ranges. At the same time, I was taking a finite elements analysis class which taught us to manually work with and construct meshes, forces, moments and boundary conditions for the analysis of loaded structures, with the addition of thermal interactions.
Nearly every useful engineering course that I took was filled with useful and complex math with emphasis in the aforementioned classes, thermodynamics, fluid mechanics, heat transfer, mechanical systems, dynamics, circuits, design optimization, mechanics of deformable bodies, and statistics. I omitted a few classes that, while math-involved, were not really difficult-math-intensive like the rest, for example my chemistry and static systems classes.
Following my graduation, I have continued to pursue a deeper understanding of the mathematical concepts that govern physics and have been particularly interested in involute curves, trochoidal curves and cycloids, as they drive nearly all gear systems. My primary source of new math concepts and relationships has been YouTube videos. There are many highly respected mathematicians running channels on YouTube who release in-depth descriptions of concepts while demonstrating them through the use of helpful visuals. Most channels that I have watched are able to present their material in an engaging way that makes the math truly exciting to viewers who may not have even heard of the concepts before. Viewers learn the math concepts, why they fundamentally operate the way that they do and their real-world applications from an actual expert on the subjects who has connections to other respected mathematicians that they frequently site in their videos.
My favorite math channel by far is 3Blue1Brown whose host also used to be the teacher in the majority of Khan Academy’s advanced math videos. Closely following 3Blue1Brown in my opinion is Minute Physics, who deals largely in real and theoretical physics concepts, often focusing on probabilistic behavior at the quantum level and extreme cases of nature’s laws. Numberphile is another greatly educational channel similar to 3Blue1Brown, though their topics tend to be more random in nature rather than interconnected. Finally, though more focused on electronics and engineering than math itself, Electroboom comprehensively covers the math involved in circuit analysis, circuit design and fundamental electrical and magnetic physics concepts. There is so much content available online made by these channels alone that there is still plenty available for me to watch and learn from these seasoned engineers, physicists and mathematicians, as well as the many other math and physics channels releasing useful content.