multivariable calculus – Second Order Change of Variables

I’m trying to simplify a Hessian computation and came across the statement in another question that a second order change of variables only works for linear changes. I tried looking in some differential geometry books and found in “Comprehensive Introduction to Differential Geometry Vol. 2” on page 201…

“Our Hessian is defined even at points were $f_astneq 0$ because we are working with a vector space and identifying it with its tangent space at $v$; this amounts to saying we are only considering linear changes of coordinates, all of which leave the quantity defined by this formula invariant.”

Why isn’t there a nonlinear version of the change of variables? With sufficient local information about a function, it seems we should be able to construct the best quadratic approximation. Are there some conditions we can impose to allow for a theorem?

Thanks!