What does f_{xy} mean?

Assume we have a function f(x,y) of two variables like f(x,y) = x^{2} y.
The partial derivative f_{x} is the rate of change of the
function f in the x direction. We also can see that _{xx} means:
it is positive if the surface is bent concave up in the x direction and negative
if it is bent concave down in the x direction.
What it the interpretation of the derivative f_{xy}?
One can interpret it as the rate of change of the slope in the xdirection
as one moves into the y direction.

Can we read of the value of
f
_{xy} from a level curves?
For one of the following contour maps, the derivative f
_{xy} is
nonzero at the center of the picture. For which one?