Globally Univalent C1-Maps with Separability.

摘要

When solving one equation in one unknown, f(x) = q, it is obvious geometrically that if f(x) is continuously differentiable and f'(x) not equal 0 for all x, then for each q the equation has at most one solution (f is then said to be univalent). Of course the univalence of f does not ensure the existence of a solution; for example, e to the x power = 0. When solving a system of n equations in n unknowns, f sub i(x1,...,xn) = q sub i (i = 1,...,n), the analogue of f'(x) is the n x n Jacobian matrix (del f sub i/del x sub k). It is interesting to investigate conditions on the Jacobian matrix which will ensure the inivalence of the left hand of the equation. Such conditions are of practical importance if they are combined with conditions which ensure the existence of solutions because if the equation has a solution and if the left hand of the equation is univalent then the solution is unique.