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.