# An example for a mathematical object that has two indexes, and is not a tensor

I understand how to check if something is a tensor (if it transforms like a tensor, it's a tensor).

I have this example of a rank 2 tensor: $$\tau_{ij}=r_if_j - r_jf_i$$ and I understand that it is a tensor because $$\sum_j\tau_{ij}e_j$$ is a vector (e is some vector).

What I'm trying to find is a mathematical "creature" with two indexes (similar to $\tau_{ij}$) that is not a tensor. (I'm just trying to get a better understanding of what is a tensor).

• Could you clarify what definition of a tensor you are familiar with? Mar 9 '16 at 11:48

A typical example of an object that fails to behave like a tensor is the $\Gamma_{ij}^k$. If you prefer to have only two indices just contract one of them and use $\Gamma_{ij}^k v^i$ where $v$ is a vector.
If you have a manifold $M$ covered with local coordinate systems $(U_\alpha,\phi_\alpha)_{\alpha\in I}$ then the transition maps $$\psi_{\beta\alpha}:=\phi_\beta\circ\phi_\alpha^{-1}$$ are identified by two indices, but have no tensorial structure.