This post is revised by Chat GPT. This post presents a detailed walkthrough of the continuity equation from physical intuition to mathematical formalism. We aim to derive the equation step by step and explain all quantities involved. 1. Motivation: Conservation in Physical Systems Many physical quantities are conserved in nature. For instance: Mass in a closed system Charge in electromagnetic systems Energy in isolated systems Number of particles in fluid dynamics ...
This post introduces the Gaussian integral, starting from basic integration and gradually building towards a probabilistic and linear algebraic perspective. The exposition proceeds from a purely mathematical derivation to a more intuitive understanding grounded in probability and linear algebra. I. The Fundamental Theorem The most fundamental Gaussian Integral is as follows: $$ \int_{-\infty}^{\infty} \exp \left(-\frac{1}{2}x^2 \right) dx = \sqrt{2\pi} $$ Equivalently, in the form of the standard Gaussian distribution: ...
Thanks to 子预. His article helped me a lot in understanding this topic. This is a brief (but not rigorous) proof of the Central Limit Theorem (CLT). The CLT is one of the most fundamental results in probability theory, showing that the sum of many independent random variables tends toward a normal distribution, regardless of their original distribution. Intuitively, the CLT explains why many natural phenomena follow the bell curve: individual randomness averages out into a predictable pattern. ...