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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).

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    $\begingroup$ Could you clarify what definition of a tensor you are familiar with? $\endgroup$ Mar 9 '16 at 11:48
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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.

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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.

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  • $\begingroup$ Nice answer, but I suspect that this answer and the one using Christoffel symbols above are a bit too advanced for the asker. $\endgroup$ Mar 12 '16 at 7:08

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