Δ (capital delta): change, like Δt = t2 - t1.
d (lowercase d): ‘infinitesimal’ change, shows up in derivatives like dy/dx and plays well with limits. More formally, tied to differential forms.
∂ (curly d, aka partial derivative) is… essentially the fancy cousin of d. Usually seen in multivariable calculus, where stuff gets too complicated and you have to hold some variables constant while you poke at others.
δ (lowercase delta) wears many hats. In the calculus of variations, it’s used to represent a variation: a tiny tweak in a function’s form, not just its value. For example, is used in the principle of least action in physics. This is not the same as a differential (like d) or a finite change (like Δ), but rather a virtual displacement. In thermodynamics, δ shows up in equations like dU = δQ + δW, where it’s used for imperfect differentials, because heat and work depend on the path taken, not just the endpoints. In real analysis, it’s the δ in epsilon-delta proofs. And then there’s the Dirac’s delta function (δ(x)), which is not a function in the traditional sense but a mathematical object that’s zero everywhere except at one point and integrates to 1. Also worth noting: δ sometimes shows up in functional derivatives, and then in completely different contexts like the Kronecker delta, which is basically a 1-or-0 identity switch.