What is not a Cartesian tensor?

Question

This is my understanding of a Cartesian tensor: Any vector in three-dimensional space can be written in a Cartesian system as $$\vec{A}=A^1\hat{e}_1+A^2\hat{e}_2+A^3\hat{e}_3$$ where $A^1, A^2, A^3$ are the components of $\vec{A}$ in a Cartesian basis $\hat{e}_1, \hat{e}_2, \hat{e}_3$. Under a rotation of the coordinate system, the components transform according to $$A^{\prime i}=\frac{\partial x^{\prime i}}{\partial x^j}A^j=R^i_{~ j} A^j$$ where $R$ is the orthogonal rotation matrix. This is a Cartesian vector.

What is not a Cartesian vector (or tensor, in general)?

References:

https://en.wikipedia.org/wiki/Cartesian_tensor

https://mathworld.wolfram.com/CartesianTensor.html

I will expand the question after some time and will also give textbook references. I am also thoroughly confused by the terminology but it does exist.

Answer 1

It's meaningless, or wrong, to define a "Cartesian tensor" or a "Cartesian vector". You can define the components of a tensor or a vector in a Cartesian reference fame, or coordinate systems if you're dealing with tensor fields.

A tensor, or a vector, is an implicit mathematical object, i.e. it doesn't depend on the reference frames you can use to write it. It's the most important property of this mathematical objects to be used in Physics, translating the independence of the physical process, i.e. the nature, from the observer, i.e. the reference frames or the coordinates used to describe it.

Answer 2

According to Synge & Schild: TENSOR CALCULUS, pp:127-128

[...] define as Cartesian tensors quantities which transform according to the same laws when the coordinates undergo an orthogonal transformation, i.e. when we pass from one set of homogeneous coordinates to another.

and then

We use the word "Cartesian" because homogeneous coordinates in a flat space are analogous to rectangular Cartesian coordinates in a Euclidean plane or 3-space, and indeed reduce to rectangular Cartesians when the space reduces to the Euclidean plane or 3-space. In order to qualify as a tensor, a set of quantities has to satisfy certain laws of transformation when the coordinates undergo a general transformation. This is a much more stringent condition than that which we impose on a set of quantities in order that it may qualify as a Cartesian tensor; in the latter case only orthogonal transformations are involved. Consequently, every tensor is a Cartesian tensor, but the converse is not true. (This is an abuse of language comparable to the following: "All horses are black horses, but the converse is not true." To avoid it, we might call the tensors of Chapter I "complete tensors." A simpler plan is to regard the expression "Cartesian tensor" as a single noun, not divisible into "Cartesian" and "tensor.")

where a homogeneous coordinate transformation is defined as $$z'_m = A_{mn}z_n + A_m \\ A_{mp}A_{nq} = \delta_{pq}$$

By this definition a Cartesian tensor is more general than a "normal" tensor.