Can I write $\frac {\partial E_{rs}} {\partial E_{mn}} = \delta_{rsmn}?$

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Can I write $\frac {\partial E_{rs}} {\partial E_{mn}} = \delta_{rsmn}?$ and then use that delta to change variables in another tensor like:

$$C_{ijkl}\delta_{rskl} = C_{ijrs}$$

or is there some other form of the delta. I wikipedia'd the generalized kronecker delta but I don't think it corresponds to the partial derivative above. I'm operating off of the fact that

$$\frac {\partial x_{i}} {\partial x_{m}} = \delta_{im}$$