TITLE: Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?
QUESTION [7 upvotes]: Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, 
$$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$
Called the "energy" of $f$? I am sure it comes from physics but I do not know why.

REPLY [4 votes]: Let us imagine that the graph of $f$ in $\mathbb{R}^3$ represents some sheet of uniform elastic rubber. If we study the idealized case, we find that the force in the horizontal direction is $\Delta f$. (This is reasonable, as when we study Laplace's equation we find that harmonic functions, those which satisfy $\Delta f =0$ satisfy a property where $f(x)$ is equal to the average of value of $f$ on any circle about $f(x)$, so the rubber will be at an equilibrium). Thus the integral given captures the potential energy in the rubber pulling on itself (you get a factor it squared so that kinetic and potential energy interact correctly).