# Little confused with tensor index manipulation

1. Feb 16, 2015

### kau

I am making mess of the following expression..
i have following expression $\frac{\partial{g}}{\partial{g_{\mu j}}} *g_{\nu j}=g \delta^{\mu} _{\nu}$
then I have sum over j only in the above expression.
But above expression is nonzero only when ${\mu}$ is equal to $\nu$.
So we have $\frac{\partial{g}}{\partial{g_{\mu j}}} *g_{\mu j}=g$ ....(a)
I can understand that there is no $\mu$ dependence in right hand side so we have to sum over ${\mu}$ also. But doing the mathematical step just summing over ${\nu}$ gives equation in ${\mu}$ but in addition how the summation in that implied????
Also now if I take partial derivative of the above expression (a) how it gives
$\frac{\partial{g}}{\partial{g_{\mu j}}} *\frac{\partial{g_{\mu j}}}{\partial{x^{i}}}=\frac {\partial{g}}{\partial{x^{i}}}$
?? why the term like $\frac{\partial^{2}{g}}{{\partial{g_{\mu j}}}{\partial{x^{i}}}}$ vanishes???
please tell me what I am missing??

2. Feb 16, 2015

### mathman

I believe you could clear things up for yourself if you explicitly used the sum sign rather than relying on the convention.