text: Assume that the phase space of the underlying dynamical system is the material configuration space of a continuum, such as a fluid or a deformable body. For instance, for a dynamical system generated by an unsteady velocity field
  the open set  of possible particle positions is a material configuration space. In this space, LCSs are material surfaces, formed by trajectories. Whether or not a material trajectory is contained in an LCS is a property that is independent of the choice of coordinates, and hence cannot depend of the observer. As a consequence, LCSs are subject to the basic objectivity (material frame-indifference) requirement of continuum mechanics. The objectivity of LCSs requires them to be invariant with respect to all possible observer changes, i.e., linear coordinate changes of the form
  where  is the vector of the transformed coordinates;  is an arbitrary  proper orthogonal matrix representing time-dependent rotations; and  is an arbitrary -dimensional vector representing time-dependent translations. As a consequence, any self-consistent LCS definition or criterion should be expressible in terms of quantities that are frame-invariant. For instance, the strain rate  and the spin tensor  defined as
  transform under Euclidean changes of frame into the quantities
  A Euclidean frame change is, therefore, equivalent to a similarity transform for , and hence an LCS approach depending only on the eigenvalues and eigenvectors of  is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of  is generally not frame-invariant.
  A number of frame-dependent quantities, such as , , , as well as the averages or eigenvalues of these quantities, are routinely used in heuristic LCS detection. While such quantities may effectively mark features of the instantaneous velocity field , the ability of these quantities to capture material mixing, transport, and coherence is limited and a priori unknown in any given frame. As an example, consider the linear unsteady fluid particle motion
  which is an exact solution of the two-dimensional Navier–Stokes equations. The (frame-dependent) Okubo-Weiss criterion classifies the whole domain in this flow as elliptic (vortical) because  holds, with  referring to the Euclidean matrix norm. As seen in Fig. 3, however, trajectories grow exponentially along a rotating line and shrink exponentially along another rotating line. In material terms, therefore, the flow is hyperbolic (saddle-type) in any frame.
  Since Newton’s equation for particle motion and the Navier–Stokes equations for fluid motion are well known to be frame-dependent, it might first seem counterintuitive to require frame-invariance for LCSs, which are composed of solutions of these frame-dependent equations. Recall, however, that the Newton and Navier–Stokes equations represent objective physical principles for material particle trajectories. As long as correctly transformed from one frame to the other, these equations generate physically the same material trajectories in the new frame. In fact, we decide how to transform the equations of motion from an -frame to a -frame through a coordinate change  precisely by upholding that trajectories are mapped into trajectories, i.e., by requiring  to hold for all times. Temporal differentiation of this identity and substitution into the original equation in the -frame then yields the transformed equation in the -frame. While this process adds new terms (inertial forces) to the equations of motion, these inertial terms arise precisely to ensure the invariance of material trajectories. Fully composed of material trajectories, LCSs remain invariant in the transformed equation of motion defined in the -frame of reference. Consequently, any self-consistent LCS definition or detection method must also be frame-invariant.