Question

What is the graphical meaning of the second order partial derivative Fxy(x0,y0)? It makes sense that Fxx(x0,y0) is the concavity of 2d graph that is formed when you slice the graph F(x,y) with a plane parallel to the x axis at the point (x0,y0). And it makes sense that Fyy is the concavity in the y direction. But Fxy doesn't make sense to me.

Answer 1

@wio

Answer 2

take them as projections or contours and will be easier to understand the geometrical meaning of higher order partial differentiation

Answer 3

also extension is the Jacobian and Hessian matrices that tell you the direction of change of function along each dimension.

Answer 4

when you say "concavity in 2D", it is the derivative of a derivative which you can measure with respect to the same variable or any of the other ones.