Upper-convected time derivative
In continuum mechanics, including fluid dynamics, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, is the rate of change of some tensor property of a small parcel of fluid that is written in the coordinate system rotating and stretching with the fluid. The operator is specified by the following formula: where: The formula can be rewritten as: By definition the upper-convected time derivative of the Finger tensor is always zero. It can be shown that the upper-convected time derivative of a spacelike vector field is just its Lie derivative  by the velocity field of the continuum. The upper-convected derivative is widely use in polymer rheology for the description of behavior of a viscoelastic fluid under large deformations. Examples for the symmetric tensor "A". Simple shear. For the case of simple shear: Thus, Uniaxial extension of incompressible fluid. In this case a material is stretched in the direction X and compresses in the directions Y and Z, so to keep volume constant. The gradients of velocity are: Thus,