Locally unique solutions

Here is an example of a system of equations whose solutions are locally unique; i.e. near a given solution there are no other solutions.

To determine that solutions of the given equation are locally unique, we used detailed knowledge of the polar coordinate transformation. Fortunately, there is an easier way: The inverse function theorem allows us to conclude that solutions to a given equation are locally unique simply by checking whether, at each solution, the derivative of the given function is an isomorphism (has non-zero determinant).

Return to the main document on the implicit function theorem.