text: Heuristically, one may seek initial positions  of repelling LCSs as set of initial conditions at which infinitesimal perturbations to trajectories starting from  grow locally at the highest rate relative to trajectories starting off of . The heuristic element here is that instead of constructing a highly repelling material surface, one simply seeks points of large particle separation. Such a separation may well be due to strong shear along the set of points so identified; this set is not at all guaranteed to exert any normal repulsion on nearby trajectories.
  The growth of an infinitesimal perturbation  along a trajectory  is governed by the flow map gradient . Let  be a small perturbation to the initial condition , with , and with  denoting an arbitrary unit vector in . This perturbation generally grows along the trajectory  into the perturbation vector . Then the maximum relative stretching of infinitesimal perturbations at the point  can be computed as
  where  denotes the right Cauchy–Green strain tensor. One then concludes that the maximum relative stretching experienced along a trajectory starting from  is just . As this relative stretching tends to grow rapidly, it is more convenient to work with its growth exponent , which is then precisely the finite-time Lyapunov exponent (FTLE)
  Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces (or ridges) of the FTLE field.
  This expectation turns out to be justified in the majority of cases: time  positions of repelling LCSs are marked by ridges of . By applying the same argument in backward time,
  we obtain that time  positions of attracting LCSs are marked by ridges of the backward FTLE field .
  The classic way of computing Lyapunov exponents is solving a linear differential equation for the linearized flow map . A more expedient approach is to compute the FTLE field from a simple finite-difference approximation to the deformation gradient.
  For example, in a three-dimenisonal flow, we launch a trajectory  from any element  of a grid of initial conditions. Using the coordinate representation  for the evolving trajectory , we approximate the gradient of the flow map as
  with a small vector  pointing in the  coordinate direction. For two-dimensional flows, only the first  minor matrix of the above matrix is relevant.