What can/do mixed second-order partial derivatives represent?

Question

What I know from basic calculus: The first derivative of a function $f:\mathbb{R}\to\mathbb{R}$ gives an indication of the function's behavior (namely, how the value of $f(x)$ changes as $x$ changes). The second derivative admits information about the concavity of the function.

On this page, I read that first order partial derivatives of a function $f:\mathbb{R^2}\mapsto\mathbb{R}$ can show the same principle; that is, $\dfrac{\partial f}{\partial x_i}$ can be thought of the rate of change of a particular cross-section of $f(x_1,x_2)$ in the direction of $x_i$. If this is the case, then it makes sense to me that $\dfrac{\partial^2f}{\partial {x_i}^2}$ is useful in showing how the concavity of a function changes in the direction of $x_i$.

So what might $\dfrac{\partial^2f}{\partial x_1\,\partial x_2}$ and $\dfrac{\partial^2f}{\partial x_2\,\partial x_1}$ represent? Do they offer anything "meaningful" in the same way that the first- and unmixed second-order partial derivatives do?

Answer 1

They are part of the Hessian matrix: $$ H = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\partial x_2} \\ \frac{\partial^2 f}{\partial x_1\partial x_2} & \frac{\partial^2 f}{\partial x_2^2} \end{bmatrix} .$$ If you want to know the concavity of $f$ in the direction $\vec a = [ a_1, a_2]^T$ you compute the quantity $$ \vec a^T H \vec a = a_1^2 \frac{\partial^2 f}{\partial x_1^2} + 2 a_1 a_2 \frac{\partial^2 f}{\partial x_1\partial x_2} + a_2^2 \frac{\partial^2 f}{\partial x_2^2},$$ and if this is positive then $f$ is concave in that direction.

You can get even more information about the concavity of $f$ as follows: if the determinant of $H$ is positive, and the trace of $H$ is positive, then $f$ is concave in all directions at that point; if the determinant of $H$ is positive, and the trace of $H$ is negative, then $f$ is convex in all directions at that point; and if the determinant of $H$ is negative, then $f$ is concave in some directions and convex in other directions. (If the determinant of $H$ is zero, then its like the second derivative being zero, and most anything can happen.)

In this manner you can develop a second derivative test for functions $f:\mathbb R^2 \to \mathbb R$ which will identify stationary values as minima, maxima or saddle points.

So while the mixed derivatives by themselves may seem at first to be of not much use, when seen as part of the bigger picture of the Hessian matrix, they become indispensable.