a harmonic function
A harmonic function is the gold-standard of a perfectly balanced, tension-free field. Stretch a soap film across a wire loop and let it settle: its height is harmonic — every point sits exactly at the average height of the ring of points around it, so the film has no bumps it could relax away. Equilibrium temperatures, electrostatic potentials in empty regions, and the heights of stretched membranes are all harmonic functions.
Precisely, a function u is harmonic on a region if it is twice continuously differentiable there and satisfies Laplace's equation, Laplacian u = 0. An astonishing fact makes the definition richer than it looks: this single equation forces the mean-value property — the value of u at any point equals its average over every sphere (and every ball) centred there that fits inside the region. The two statements are equivalent. From them flow a cascade of consequences: harmonic functions are automatically infinitely differentiable, indeed real-analytic, even if you only assumed two derivatives; they obey the maximum principle (no interior maxima or minima); and they are extremely rigid — knowing u on a boundary determines it everywhere inside.
In the plane there is a beautiful bonus: the real and imaginary parts of any complex-analytic function are harmonic, and conversely every harmonic function is locally the real part of an analytic one. So planar potential theory and complex analysis are two faces of the same subject. A common misconception is that harmonic means oscillating like a sine wave; the name comes from the harmonics of a vibrating membrane, but a harmonic function itself is the smooth, non-oscillating equilibrium — it has no wiggle room at all.
In two dimensions, u(x, y) = x^2 - y^2 is harmonic: u_xx = 2 and u_yy = -2, so u_xx + u_yy = 0. So is u = x y, and u = e^x cos y, and u = log(x^2 + y^2) away from the origin. Each is the real or imaginary part of an analytic function (z^2, z^2/2, e^z, log z).
Quick test: add the unmixed second derivatives; if they cancel, the function is harmonic.
Harmonic does not mean small or bounded — log(x^2+y^2) blows up at the origin, and x^2-y^2 grows without bound. The defining property is local balance (Laplacian u = 0), not size. And in one dimension harmonic just means u_xx = 0, i.e. a straight line — the rich theory needs two or more dimensions.