It's also easier to motivate them (and get an intuition for them) if you have a physical motivation for them. Like if you imagine being in Fourier's shoes (or H

It's also easier to motivate them (and get an intuition for them) if you have a physical motivation for them. Like if you imagine being in Fourier's shoes (or Heaviside's) and needing to come up with techniques for solving differential equations more easily. Fourier series let you solve PDEs with linear algebra, and Laplace transforms turn ODEs into polynomial equations, both of which are easier to deal with. Or to put it another way, if you imagine being faced with diff eqs and thinking "Hmm, is there a way I can turn these into problems I already know how to solve... like, say, polynomials?", then the idea that there even exists an invertible transformation that lets you do such a thing is itself quite fascinating in its own right—regardless of any elegance the transformation itself might have from a theoretical mathematical standpoint. And having such a motivation and application in mind (and knowing the physical nature of the problem, where we know we'll end up with waves) helps ground the idea in something very concrete and avoids making it look like it's just a random transformation we're studying because we're bored.