Let M be a manifold and let f: m -> R a Morse function. Let x be a critical point of f and assume all critical points are non-degenerate. Let W^u(x) be th unstable manifold of x when considering the negative gradient flow on M. Why does the tangent space at x to W^u(x) = Eig^- H^2f(x)? Denote the Hessian by H^2f(x). I know that since the critical points are non-degenerate the Morse lemma gives a sort of quadratic decomposition of f. I also know the M can be written as the union over all x of the W^u(x). One of the problems is that I don't really understand the object T_xW^u(x). The tangent space is a vector space. So T_xW^u(x) consist of the points in the vector space that "repel" from x?