I'm trying to better understand exactly what $f_{xy}(x,y)$ at a point is geometrically, and possibly understand why $f_{xy}$ and $f_{yx}$ should be equivalent, not just because the math happened to make it so. For example, $f_{xx}$ would be like looking at the concavity of the function in only the $x$ direction.
My real question is about how the second partial derivative test works.
$$D(x,y)=f_{xx}(x,y)f_{yy}(x,y)-(f_{xy}(x,y))^2$$
I can see that the sign of the first term can be interpreted as whether or not the concavity in both the $x$ and $y$ directions are in the same direction. For a critical point $(a,b)$, if the concavities are opposite, then we have a saddle point. It's also easy to see that $D(a,b)<0$.
When the concavities do agree, then $f_{xy}$ starts to play a role in the sign of $D$. My current understanding on why this is necessary, is because looking at the second derivatives at only the $x$ and $y$ directions doesn't quite give the entire picture on what is happening at $(a,b)$, but including $f_{xy}$ gives sufficient information to determine if $(a,b)$ is truly a local extremum or a saddle point instead (given that $D\neq0$).
My problem is that I still don't exactly get what sort of information $f_{xy}$ entails for a given point.
