This is not advanced, but Rodrigues' rotation formula (https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula https://en.wikipedia.org/wiki/Rodrigues%27_ro

This is not advanced, but Rodrigues' rotation formula (https://en.wikipedia.org/wiki/Rodrigues%27_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.