Abstract: The main purpose of the present paper is to obtain the relation between the abstract presentation of Euclidean objectivity and its component representation with respect to the general curvilinear coordinate systems with a relative rigid motion under a fixed frame of reference, so that the tensor representation of Euclidean objective quantity has its component form. This relation is based on the fact that an objective quantity under the frame of reference with respect to the fixed coordinate system of frame of reference has the same component with the objective quantity under the frame of reference relative to another coordinate system of with the same relative rigid motion as that of and to the fixed coordinate system of . On the other hand, it is exemplified in the present paper that the principle of material frame – indifference with the admissible equivalent dynamic processes of a body by Truesdell and Noll (1965) 14 is more convenient than the form-invariance of constitutive functionals by Svendsen and Bertram (1999) 13 and Liu (2004)4, especially for the rate-form constitutive equations.