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So, I got my first orders, e.g.:

dF/du = dF/dx*dx/du + dF/dy*dy/du + dF/dz*dz/du

Defining X=[x y z] and U=[u v] I can formulate this in vector notation:

dF/dU = dF/dX * Jacobian(X(u,v))

at least I think I can. It seems to be working.

Now I need the second orders of F with respect to [u,v]. What I really need is the 2x2 Hessian matrix. I'm not totally sure how to proceed. I plowed through and got all my partials of F with respect to [x,y,z], but I'm not sure how to apply the chain rule or its equivalent either in scalar or matrix/vector notations.

Can anyone help? (If nothing else, how do you write out ddF/dudv in terms of partials of F(x,y,z) and G(u,v)?)