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.

 

[blue_raver22]

The geometric interpretation for ##f_{xy}(x_0, y_0)## is the rate of change of the slope in the x-direction with respect to the y-direction at a specific point (x0, y0) in the function f(x,y). This can be visualized as the curvature of the surface formed by the function in the x-y plane at that point. In other words, it represents how the slope of the function changes as we move along the y-axis at a fixed x-value. This concept is important in understanding the behavior of surfaces and their derivatives in multivariate calculus and has practical applications in fields such as engineering and physics.