Worked example - operators on a central field. Take f(r) = 1/r (Coulomb potential shape), r = sqrt(x^2+y^2+z^2). Step 1: grad f = df/dr r-hat = (-1/r^2) r-hat = -r_vec/r^3. In components: df/dx = -x/r^3 etc. Step 2: div(grad f) = nabla^2(1/r): compute div(-r_vec/r^3) = -(3/r^3) + 3(r_vec.r_vec)/r^5 = -(3/r^3) + 3/r^3 = 0 for r != 0. So nabla^2(1/r) = 0 away from the origin - Laplace's equation holds in free space, as electrostatics demands. Step 3: at the origin the singularity gives nabla^2(1/r) = -4 pi delta^3(r) - the point charge, previewing the Dirac delta. Also curl(grad f) = 0 confirms E = -grad V is irrotational in electrostatics.