from Which Differential Equations are Easy to Solve? by John Starrett
Posts tagged with ODE's
In harmonic analysis and PDE, one often wants to place a function
ƒ:ℝᵈ→ℂon some domain (let’s take a Euclidean space
ℝᵈfor simplicity) in one or more function spaces in order to quantify its “size”….
[T]here is an entire zoo of function spaces one could consider, and it can be difficult at first to see how they are organised with respect to each other.
For function spaces
Xon Euclidean space, two such exponents are the regularity
sof the space, and the integrability
pof the space.
Hat tip: @AnalysisFact
Some level surfaces (isoclines) of the simplest atom’s electron position. (Electron orbitals.)
Remember that electrons control chemistry i.e. why things are the way they are on Earth. (High-energy physics, like as high of energy as a star, is where the new particles and quantum gravity, QCD, and such take place.)
Standing waves in 2-D via dhiyamuhammad.
Pretty amazing that if you simply add together oscillations = vibrations = waves = harmonics and constrain them within a box, that all these shapes emerge. (See this video for such waves being constructed in real life). By the way, mathematicians sometimes refer to these as “square drumhead” problems because a drumhead is a real-life 2-D surface that vibrates in exactly these kinds of ways to produce the sounds we associate with various drums.
In the link Muhammad points to—Harmonic Resonance Theory—the mathematics of standing waves are applied to the problem of Gestalt in psychology of sense experience.