I Quotient law and the curl in index notation

[SiennaTheGr8]

TL;DR Summary

In index notation, the curl can be expressed in a way where the quotient law would seem to "fail." There must be a subtlety that I'm missing.

If I'm not mistaken, the curl can be expressed like this in index notation:

$(\nabla \times \vec v)^i = \epsilon^{i j k} \nabla_j v_k = \epsilon^{i j k} (\partial_j v_k - \Gamma^m_{j k} v_m) = \epsilon^{i j k} \partial_j v_k$

(where the last equality is because $\epsilon^{i j k}\Gamma^m_{j k}$ is both symmetric and anti-symmetric in $j$ and $k$). But now with $(\nabla \times \vec v)^i = \epsilon^{i j k} \partial_j v_k$, doesn't the quotient law say that $\partial_j v_k$ is a tensor, even though it obviously isn't? Clearly I'm wrong. Is there a subtlety to the quotient law that I'm missing?

 

[anuttarasammyak]

Quotient law is not applicable here because $\epsilon$ is not an arbitrary tensor.