Cartesian tensor
From Glossary of Meteorology
Cartesian tensor
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.]
Copyright 2022 American Meteorological Society (AMS). For permission to reuse any portion of this work, please contact permissions@ametsoc.org. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act (17 U.S. Code § 107) or that satisfies the conditions specified in Section 108 of the U.S.Copyright Act (17 USC § 108) does not require AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a website or in a searchable database, or other uses of this material, except as exempted by the above statement, require written permission or a license from AMS. Additional details are provided in the AMS Copyright Policy statement.
