Spherical Harmonics and the Multipole Expansion

In this video, we explore the mathematical beauty of spherical harmonics: special functions that form the natural basis for patterns on a sphere. From quantum mechanics to cosmology, spherical harmonics help us describe complex systems in a compact and elegant way.

We'll start by understanding how functions on a sphere can be decomposed into simple components and then build toward the multipole expansion—a method for expressing fields like those of gravity or electrostatics in terms of their angular structure.

Topics include:

Visual intuition for spherical harmonics
Solution of the Laplace equation in spherical coordinates
Physical examples from gravitational and electromagnetic fields to the structure of the Cosmic Microwave Background
The meaning of monopoles, dipoles, quadrupoles, and beyond

Prerequisites for this video are a basic understanding of multivariable calculus, (partial) differential equations and the Fourier series.

Whether you're a physics student or just curious about the mathematics of nature, this is your crash course into one of the most elegant tools in theoretical physics.

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Videos about the Fourier Series:
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Other Resources:
https://www.g-red.eu/geoid/geoidViewe...
https://icgem.gfz-potsdam.de/vis3d/lo...

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Credits:

This video was created for the Summer of Math Exposition 2025
Made with Manim Community Edition v0.19.0

European Space Agency - ESA
GFZ Helmholtz-Zentrum für Geoforschung
CAMB (Code for Anisotropies in the Microwave Background)

Music by Chris Zabriskie
Tracks:
I Am Running Down the Long Hallway of Viewmont Elementary
John Stockton Slow Drag
Short Song 012023
Cut to Rip Torn
Licensed under Creative Commons Attribution 4.0