Symmetry in Physical Laws
Symmetry in physics isn't just about visual balance. It means an operation leaves physical laws unchanged. I explain the deep…
Study Notes
Macrostates and Microstates and their relations with Thermodynamic Probability
The difference between macrostates and microstates confused me for months in statistical physics. Then a concrete example made everything click.…
How to change Differential Equations into Integral Equations?
Every differential equation can be converted into an integral equation. Just integrate it. Sounds too simple, right? The details matter…
Square Integrable Functions, Norms, Trial Method in Integral Equations
Building on integral equation fundamentals, this article covers square integrable functions, norms, and the trial method for solving integral equations.…
Definitions and Types of Integral Equations
Integral equations look intimidating, but they're built from ideas you already know. I explain what integral equations are, their classification…
Examination Strategies : Tactics & Tips
Exam preparation isn't just about studying harder. It's about studying smarter. I cover practical strategies for managing your nervous system,…
Complete Elementary Analysis of Ramanujan’s Nested radicals
I wrote this analysis in 2010 at age 17, obsessed with Ramanujan's nested radicals while studying under the INSPIRE-SHE scholarship.…
Real Sequences: Definitions, Theorems, and Examples
Real sequences form the backbone of real analysis and advanced calculus. I've taught this topic to dozens of students, and…
Euler’s (Prime to) Prime Generating Equation
Euler discovered a simple formula that generates 40 consecutive prime numbers: n squared plus n plus 41. Plug in 0…
Irrational Numbers and The Proofs of their Irrationality
How do you prove a number is irrational? You can't check infinitely many decimal places. You need a proof. I…
The Area of a Disk
The area of a disk is pi times radius squared. You memorized this in middle school. But do you know…
Triangle Inequality
The triangle inequality sounds obvious. No side of a triangle can be longer than the other two sides combined. But…