I’m back! I decided to blog this because it didn’t feel substantial enough to have notes on its own and I couldn’t find a proper place for it elsewhere so here I am. Today, I wanted to discuss natural units i.e. the system under which , , etc. It became apparent to me that manyContinue reading “Natural Units”
Often, the first exposure one gets to rigorous mathematics is the definition of a limit. Let’s consider what this is for a sequence. We say if This, at first sight, is ugly. It takes a while to even understand what it’s saying, longer to see why it works, and much longer to apply it. It’sContinue reading “A Natural Limit Definition”
Quantum Mechanics since its inception has been one of the most philosophically controversial concepts in all of physics. But what really is so confusing about quantum mechanics? The answer lies in two fundamental principles: locality and realism. Locality – locality asserts that all information and matter in the universe is limited by the speed of light. NoContinue reading “Philosophy of Quantum Mechanics”
It has admittedly been quite a while since my last post over a year ago. I thought I would restart the posts by revisiting one of the first topics I discussed on the website: quaternions. My previous post, upon review, seems to be quite uninformative on what the nature and use of them are whichContinue reading “Quaternions Revisited”
In the first part, we discussed the idea of a functional, what it means, and how to find its extrema using the calculus of variations. However, those equations don’t really capture how amazing and applicable calculus of variations really is so the following will be some examples of this. In fact, the drawn out results fromContinue reading “Calculus of Variations Part 2: Lines, Bubbles, and Lagrange”
In the first post, we established a general intuition of how forms work and why they may provide a better geometric intuition of what is actually occurring. It was mentioned that these ideas extend the ideas of vector calculus so it seems natural to see how differential operators like gradient, curl, and divergence arise inContinue reading “Differential Forms Part 2: Differential Operators and Stokes Theorem”
Math is cool. Here are some cool things in math that I don’t think are really extensive enough for their own post but I still want to share. 1. The following is a great reason why.
Calculus of variations is an extremely useful and amazing tool in physics, math, computer science, and a variety of fields. Similar to how regular calculus is focused around functions and differentials, this field focuses on functionals and variations. A functional takes in a function and spits out a number. The following are examples of functionals.
The idea of infinity is easy to look over upon first glance. It can simply be defined as the idea that numbers go on forever and it is easy to end there. However, there are very developed and well-defined notions of infinity suggest that there are different orders and types of infinity which come withContinue reading “Orders of ∞”
Differential forms is a topic that, in some sense, extends ideas presented in vector calculus with more suggestive notation and geometric intuition into higher dimensions. The distinction may seem small and insignificant especially in the third dimension that we live in but its results and implications are quite elegant and can lead to nice formalizationContinue reading “Differential Forms Part 1: Dimensions and Notation”