Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior and to determine the associated manifold structure in the flow. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting, defined by finite-time Lyapunov exponents and vectors, is interpreted in relation to the asymptotic theory of partially hyperbolic sets. The finite-time Lyapunov approach relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. Examples of determining slow manifolds and solving Hamiltonian boundary-value problems associated with optimal control are described.