This paper discusses the application of Lyapunov theory in chaotic systems to the dynamics of tracer gradients in two-dimensional flows. The Lyapunov theory indicates that more attention should be given to the Lyapunov vector orientation. Moreover, the properties of Lyapunov vectors and exponents are explained in light of recent results on tracer gradients dynamics. Differences between the different Lyapunov vectors can be interpreted in terms of competition between the effects of effective rotation and strain. Also, the differences between backward and forward vectors give information on the local reversibility of the tracer gradient dynamics. A numerical simulation of two-dimensional turbulence serves to highlight these points and the spatial distribution of finite time Lyapunov exponents is also discussed in relation to stirring properties. (c) 2002 American Institute of Physics.