DERIVATIVES FOR THE STABLE AND UNSTABLE MANIFOLDS OF A C^r Diffeomorphism of<font>R</font>²

We illustrate how to form the Taylor polynomial for a stable or unstable manifold when considered as a curve in the plane. Our method does not use power series or assume that the manifold is analytic as is done in the method of Poincaré. Instead we use simple calculus to obtain closed form expressions for the Taylor coefficients.