Are there any geometric interpretation for the second partial derivative? i.e.
$$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$
In particular, I'm trying to understand the determinant from second partial derivative test for determining whether a critical point is a minima/maxima/saddle points:
$$D(a, b) = f_{xx}(a,b) f_{yy}(a,b) - f_{xy}(a,b)^2$$
I have no trouble understanding $f_{xx}(x,y)$ and $f_{yy}(x,y)$ as the of measure of concavity/convexity of f in the direction of x and y axis. But what does $f_{xy}(x,y)$ means?