The finite-time Lyapunov exponent (FTLE) is extensively used as a criterion to reveal fluid flow structures, including unsteady separation/attachment surfaces and vortices, in laminar and turbulent flows. However, for large and complex problems, flow structure identication demands computational methodologies that are more accurate and effective. With this objective in mind, we propose a new set of ordinary differential equations to compute the flow map, along with its first (gradient) and second order (Hessian) spatial derivatives. We show empirically that the gradient of the flow map computed in this way improves the pointwise accuracy of the FTLE field. Furthermore, the Hessian allows for simple interpolation error estimation of the flow map, and the construction of a continuous optimal and multiscale $L_p$ metric. The Lagrangian particles, or nodes, are then iteratively adapted on the flow structures revealed by this metric. It is found that Lagrangian Coherent Structures are best revealed with the minimum number of vertices with the $L_1$ metric.