The wave equation can be seen as infinitely many infinitesimal coupled oscillators by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16TODO confirm, see also: coupled oscillators. And then this idea can be used to define/motivate quantum field theory in terms of quantum harmonic oscillators with second quantization.
- youtu.be/SMmFgIEGYtw?t=324 Quantum Field Theory 2a - Field Quantization I by ViaScience (2018)
Start with: Section "String polarization".
Then go to: Section "Polarization of light".
This is about the polarization of a string in 3D space. That is the first concept of polarization you must have in mind!
Far field approximation to Kirchhoff's diffraction formula, i.e. when the plane of observation is far from the object diffracting.
Near field approximation to Kirchhoff's diffraction formula, i.e. when the plane of observation is near the object diffracting.
Notably used for the pattern of the double-slit experiment.
Shows up when trying to solve 2D wave equation on a circular domain in polar coordinates with separation of variables, where we have to decompose the initial condition in termes of a fourier-Bessel series, exactly like the Fourier series appears when solving the wave equation in linear coordinates.
For the same fundamental reasons, also appears when calculating the Schrödinger equation solution for the hydrogen atom.
Completeness: math.stackexchange.com/questions/2192665/is-this-set-of-bessel-functions-a-basis-for-all-c10-a-functions TODO
This is the Bessel function analogue to Fourier basis is complete for .
eigenvalue problem of Laplace's equation.
Existence and uniqueness of solutions of partial differential equations by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Unlike for ordinary differential equations which have the Picard–Lindelöf theorem, the existence and uniqueness of solution is not well solved for PDEs.
For example, Navier-Stokes existence and smoothness was one of the Millennium Prize Problems.
Pinned article: Introduction to the OurBigBook Project
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