Deformation of materials in the duct flow depends on the initial position and the orientation of the material element. To understand the details of the deformation process, one has to evaluate the deformation gradient tensor as a function of time and the initial position over the entire cross section in the three-dimensional duct flow. Therefore, the present paper proposes a simple method to effectively calculate the deformation gradient tensor over the entire section in the three-dimensional duct flow in a cartesian coordinate system aligned with the particle trajectory orbit. Components of the deformation gradient tensor in such a special coordinate system are found to play an important role in understanding a deformation measure in duct flows.