Hello. I have a task to find such change of variables (it should be diffeomorphism), which will transform a function to the standart Morse-type function (saddle). For example, I have a function 1/(1+x*y), which I should turn to u^2 - v^2. teacher told me, that there is a function in Wolfram Mathematica, which can do this automatically, but I can't find something like this in documentation. Please, help me. How can I solve this?
Perhaps your teacher was referring to this (or some other) demonstration:
Thank you, I saw it, but actually, it's not the thing, I'm asking about... I need the way to find the change of variables, to see this change and write it like an explicit function.
Please consider to deliver a slight bit more information to the community forum than the teacher did deliver to you.
x y = -1
f[x_,y_] := 1/(1 + x y)
then there are entries like Hessian which could be used once the task at hand is clarified.