Higher Order Partial Derivatives

A higher order partial derivative is simply a partial derivative of a partial derivative: you take the rate of change, and then ask how that rate is changing. In one variable, $f''$ told you whether a curve bends up or down, and second partials do the same job for each direction on a surface. The symbols $f_{xx}$, $f_{yy}$, $f_{xy}$ look like alphabet soup, but they just list the steps: $f_{xx}$ (differentiate in $x$ twice), $f_{yy}$ (in $y$ twice) and $f_{xy}$ (first in $x$, then in $y$). Clairaut's theorem, $f_{xy} = f_{yx}$, says the order of the two steps doesn't matter. That's most of the lesson right there.

With two variables there are four second partials, because after the first derivative you can pick either variable again. $f_{xx}$ is the bending of the $x$-slices, and $f_{yy}$ the bending of the $y$-slices. The mixed one, $f_{xy}$, is a bit stranger: it measures how the $x$-slope changes as you step in the $y$-direction, like noticing that a hillside gets steeper going east the farther north you are.

Notation. $f_{xy}$ means differentiate in $x$ first, then $y$, read left to right. In Leibniz notation the same thing is $\frac{\partial^2 f}{\partial y\,\partial x}$, read right to left, since the derivative closest to $f$ acts first.

Clairaut's theorem says $f_{xy} = f_{yx}$ whenever both are continuous, which covers everything in this course. It's not obvious that it should be true; it says "how the east-slope changes going north" equals "how the north-slope changes going east." The practical upshot is that you can choose whichever order is easier.

Second partials in physics. The heat equation $u_t = k\,u_{xx}$, the wave equation $u_{tt} = c^2u_{xx}$ and Laplace's equation $u_{xx} + u_{yy} = 0$ are all built from second partials. Functions that satisfy Laplace's equation are called harmonic; they describe steady-state temperatures and electric potentials.

The heat, wave and Laplace equations, the core of the PDE course, are equations about second partial derivatives.