From AMS Glossary
A quantity specified by components that transform according to prescribed rules under rotations of (Cartesian) coordinate axes.
A Cartesian tensor of rank zero is a scalar and is invariant under rotations. A Cartesian tensor of rank one is a vector, the components of which transform under rotations according to a single 3×3 rotation matrix. Cartesian tensors of rank two have nine components that transform according to a product of two 3×3 rotation matrices. Tensors of higher rank may be defined in similar fashion. As examples related to meteorology, mass is a scalar, velocity is a vector, and the stress tensor is a Cartesian tensor of rank two. [Because of the restriction to transformation under rotation, a Cartesian tensor need not be a (general) tensor. The latter has components that transform in a prescribed way under arbitrary changes of coordinates.]