The most beautiful applied mathematics

This is not advanced, but Rodrigues' rotation formula (https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula) which is used to act on vectors given an axis-angle vector representation of the rotation. This calculation or similar is used all the time in 3D programming, but I thought the derivation would be a bunch of unintuitive trig so I never bothered with the proof until I was bored. Given a vector v, v can be thought to act on other vectors u as (v x u). The "v x" can be represented by a rank one anti-symmetric matrix (with 3 degrees of freedom, same as v) which can be thought of as a "rotation generator". In fact, anti-symmetric matrices are the Lie algebra which generate rotation matrices. One way to compute the exponential is by a matrix Taylor series. By some algebraic properties of anti-symmetric matrices, this expanded Taylor series can easily be rearranged into Rodrigues' rotation formula.

This is on the simpler side of applied math but I think it leads well to thinking about interesting things like encodings of geometric objects (vector versus rank-constrained tensor) and to geometric algebra which is becoming more popular in game development.

Not a specific piece of applied mathematics, more a state of mind to approach and master: almost anything to do with Dimensional Analysis

https://en.wikipedia.org/wiki/Dimensional_analysis

The classic example is the derivation of the Reynolds number (mentioned in link under examples);

If you take all the things that matter about a pipe and a fluid: speed, density, length, viscosity .. and arrange those so that all dimensions cancel out you're left with a quantity that expresses a key property of the system.

This has had repeated practical applications when thinking about new systems over the past decades I've been working.

Another nugget of thought from applied mathematics was the notion that any iterated dynamic systems will end up in one of five places (subject to scaling by a constant)

-1, 0, 1, infinity, or sin()

It'll flip, diminish to nothing, stabilise, exponentiate, or wobble.

( or combinations thereof, subject to caveats, etc. YMMV )