The Multidimensional Gaussian Integral — Explained Visually

Описание: In this high-level visual lesson, we explore the Multidimensional Gaussian Integral, the mathematical backbone of probability theory, machine learning, quantum physics, and PDEs.

∫ (e^(-xTAx))dx= π^(n/2)(det(A))^-(1/2)
𝑅
𝑛


We don’t just compute it — we visualize how the matrix
A warps space and how its determinant controls the volume scaling.

This is how advanced mathematics becomes geometric, intuitive, and beautiful.

🧠 What You’ll Learn

Why Gaussian integrals are everywhere in physics

How to diagonalize symmetric matrices to simplify integrals

The geometric meaning of

det(A) as volume scaling

How this connects to the Quantum Harmonic Oscillator

The role of Gaussian integrals in Machine Learning & AI

⏱️ Timestamps

0:00 — The Ubiquity of Gaussians
1:30 — Geometric intuition: Elliptic contours of
x^TAx
3:20 — Change of variables and diagonalization
6:10 — Determinant as volume transformation
9:00 — Deriving
𝜋^𝑛/2(det(𝐴))^−1/2


11:30 — Applications in Quantum Physics
13:40 — The Gaussian in Probability & AI
15:00 — Closing Thoughts + Next Concepts

📘 Learn More

To go deeper into the analysis behind Gaussian integrals, special functions, and advanced calculus transformations — explore my book Advanced Integration Techniques.
It’s your complete roadmap to research-level understanding in math, physics, and engineering.

👉 Get the Book: https://www.amazon.ca/dp/B0F92Q89TR

🎓 Work with Me Directly: https://www.stem1online.com want to learn more? check out my channel, ‪@STEMOnlineAcademy‬