Geometric interpretation for d²f/dxdy

[Jhenrique]

If the following integral:
$$\\ \iint\limits_{a\;c}^{b\;d} f(x,y) dxdy$$ represents:

So which is the geometric interpretation for $f_{xy}(x_0, y_0)$ ?

 

[Jilang]

Would it be the rate of change of z with area, dz/dA at x0, y0?

 

[Jhenrique]

Would it be the rate of change of z with area, dz/dA at x0, y0?

Yeah! But which is the geometric interpretation?

 

[Jilang]

I am taking my last guess back. If A is the shaded area
f(x,y) = dA/dR
So df/dR is a measure of the curvature. When it is zero the area A is flat.