text: 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:
  * is the upper-convected time derivative of a tensor field 
  * is the substantive derivative
  * is the tensor of velocity derivatives for the fluid.
  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 used in polymer rheology for the description of the behavior of a viscoelastic fluid under large deformations.