Philosophy of Quantum Mechanics

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”

Calculus of Variations Part 2: Lines, Bubbles, and Lagrange

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”

Differential Forms Part 2: Differential Operators and Stokes Theorem

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”

Calculus of Variations Part 1: Establishing the Basis

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.

Differential Forms Part 1: Dimensions and Notation

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”