# Understanding the Laplace operator conceptually

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 $\Delta f (p)$ 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 $\int x^i dx$ vanish because $x^i$ is an odd function under reflection in the $x^i$ direction, and similarly the integrals $\int x^i x^j dx$ vanish whenever $i\ne j$; so this simplifies to
where $C$ is a constant depending only on the dimension.