Quotient law and the curl in index notation

  • #1
SiennaTheGr8
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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?
 
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  • #2
Quotient law is not applicable here because ##\epsilon## is not an arbitrary tensor.
 
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