text: As a simplest approach to rotational coherence, one may define an elliptic LCS as a tubular material surface along which small material volumes complete the same net rotation over the time intervall  of interest.
  A challenge in that in each material volume element, all individual material fibers (tangent vectors to trajectories) perform different rotations.
  To obtain a well-defined bulk rotation for each material element, one may employ the unique left and right polar decompositions of the flow gradient in the form
  where the proper orthogonal tensor  is called the rotation tensor and the symmetric, positive definite tensors  are called the left stretch tensor and right stretch tensor, respectively.
  Since the Cauchy–Green strain tensor can be written as
  the local material straining described by the eigenvalues and eigenvectors of  are fully captured by the singular values and singular vectors of the stretch tensors. The remaining factor in the deformation gradient is represented by , interpreted as the bulk solid-body rotation component of volume elements. In planar motions, this rotation is defined relative to the normal of the plane. In three dimensions, the rotation is defined relative to the axis defined by the eigenvector of  corresponding to its unit eigenvalue. In higher-dimensional flows, the rotation tensor cannot be viewed as a rotation about a single axis.
  In two and three dimensions, therefore, there exists a polar rotation angle (PRA)  that characterises the material rotation generated by  for a volume element centered at the initial condition . This PRA is well-defined up to multiples of . For two-dimensional flows, the PRA can be computed from the invariants of  using the formulas
  which yield a four-quadrant version of the PRA via the formula
  For three-dimensional flows, the PRA can again be computed from the invariants of  from the formulas
  where  is the Levi-Civita symbol,  is the eigenvector corresponding to the unit eigenvector of the matrix .
  The time  positions of elliptic LCSs are visualized as tubular level sets of the PRA distribution . In two-dimensions, therefore, (polar) elliptic LCSs are simply closed level curves of the PRA, which turn out to be objective. In three dimensions, (polar) elliptic LCSs are toroidal or cylindrical level surfaces of the PRA, which are, however, not objective and hence will generally change in rotating frames. Coherent Lagrangian vortex boundaries can be visualized as outermost members of nested families of elliptic LCSs. Two- and three-dimensional examples of elliptic LCS revealed by tubular level surfaces of the PRA are shown in Fig. 10a-b.