  A four-parameter kinematic model for the position of a fluid parcel in a
time-varying ellipse is introduced. For any ellipse advected by an arbitrary
linear two-dimensional flow, the rates of change of the ellipse parameters are
uniquely determined by the four parameters of the velocity gradient matrix, and
vice versa. This result, termed ellipse/flow equivalence, provides a stronger
version of the well-known result that a linear velocity field maps an ellipse
into another ellipse. Moreover, ellipse/flow equivalence is shown to be a
manifestation of Stokes' theorem. This is done by deriving a matrix-valued
extension of the classical Stokes' theorem that involves a spatial integral
over the velocity gradient tensor, thus accounting for the two strain terms in
addition to the divergence and vorticity. General expressions for various
physical properties of an elliptical ring of fluid are also derived. The
ellipse kinetic energy is found to be composed of three portions, associated
respectively with the circulation, the rate of change of the moment of inertia,
and the variance of parcel angular velocity around the ellipse. A particular
innovation is the use of four matrices, termed the IJKL basis, that greatly
facilitate the required calculations.
