The Laplace operator: those of you who now understand it, how would you explain what it “does” conceptually? How do you wish you had been taught it?
Any good essays (combining both history and conceptual understanding) on the Laplace operator, and its subsequent variations (e.g. Laplace-Bertrami) that you would highly recommend?
The Laplacian Δf(p) is the lowest-order measurement of how f deviates from f(p) “on average” – you can interpret this either probabilistically (expected change in f as you take a random walk) or geometrically (the change in the average of f over balls centred at p). To make this second interpretation precise, write the Taylor series
The integrals ∫xidx vanish because xi is an odd function under reflection in the xi direction, and similarly the integrals ∫xixjdx vanish whenever i≠j; so this simplifies to
where C is a constant depending only on the dimension.
The Laplace-Beltrami operator is essentially the same thing in the more general Riemannian setting – all the nasty curvy terms will be higher order, so the same formula should hold.