I recently come across a paper in which the notation of some equation confuses me a lot. Let's say, if I have an expression represented by delta $\delta_{jk},\delta_{jl}$, tensor $e_{jk}, e_{jl}$, and some components of a vector ($x^0_{k}, x^0_l$):
$s=(\delta_{jk}+e_{jk})x^0_k(\delta_{jl}+e_{jl})x^0_l$-----------------(1)
And if I differentiate this with respect to one component of the tensor, say $e_{\alpha \beta}$, it gives me:
$\frac{\partial s}{\partial e_{\alpha \beta}}=2(\delta_{\alpha k}+e_{\alpha k})x^0_kx^0_\beta$-----------------------(2)
How can I get (2) based on (1)? could anybody give me some details? Thank you!
Using the fact that $\frac{\partial}{\partial e_{\alpha \beta}}(e_{ij})=\delta_{i\alpha}\delta_{j\beta}$:
$$\begin{align}\frac{\partial}{\partial e_{\alpha \beta}}(s)&=\frac{\partial}{\partial e_{\alpha \beta}}(\delta_{jk}x_k^0e_{jl}x_l^0)+\frac{\partial}{\partial e_{\alpha \beta}}(e_{jk}x_k^0\delta_{jl}x_l^0)+\frac{\partial}{\partial e_{\alpha \beta}}(e_{jk}x_k^0e_{jl}x_l^0)\\&=\delta_{jk}x_k^0\delta_{\alpha j}\delta_{\beta l}x_l^0+\delta_{j\alpha}\delta_{\beta k}x_k^0\delta_{jl}x_l^0+2\delta_{\alpha j}\delta_{\beta k}x_k^0e_{jl}x_l^0\\&=x_k^0\delta_{\alpha k}x_{\beta}^0+x_\beta^0\delta_{\alpha k}x_k^0+2x_\beta^0e_{\alpha k}x_k^0\\&=2(\delta_{\alpha k}+e_{\alpha k})x_{\beta}^0x_k^0\end{align}$$