The deformation gradient in curvilinear coordinates

This short article offers an overview of the deformation gradient and its determinant in the case of curvilinear coordinates. In particular, the relation to a pure Cartesian description and the representation relative to a curvilinear basis is treated. This brief exercise is known, but hard to be found in the basic literature on Continuum Mechanics. Further, most of the literature on constitutive modeling refers to the symbol J, but J is not generally equivalent to det F. The difference is derived in detail.