I'm just learning this convention, and at times I'm a little lost, like now.
I have to calculate the following, knowing that $a_{ij}$ are constants such that $a_{ij}=a_{ji}$:
$$ \frac{\partial}{\partial x_{k}} \left[a_{ij}x_{i}\left(x_{j}\right)^{2}\right] $$
The answer I'm given I end up with:
$$ a_{ik}\left[\left(x_{i}\right)^{2}+2x_{i}x_{k}\right]$$
And this I don't understand. Why do I change to index $k$, and substitute $j$ with $i$ ? In my opinion, if I just use the product rule, I end up with:
$$ \frac{\partial}{\partial x_{k}} \left[a_{ij}x_{i}\left(x_{j}\right)^{2}\right] = a_{ij}\left[x_{i} \frac{\partial \left(x_{j}\right)^{2}}{\partial x_{k}}+ \frac{\partial x_{i}}{\partial x_{k}}\left(x_{j}\right)^{2}\right] = a_{ij}\left[\left(x_{j}\right)^{2}+2x_{i}x_{j}\right]$$
But maybe that is just as correct, or am I missing something?
Best Answer
I think this is because after differentiation your bracket gives \begin{equation} a_{ij} \left( \delta_{ki}(x_{j})^{2}+2x_{i} \delta_{kj} x_{j}\right) \end{equation} Allowing $k \rightarrow j$ \begin{equation} a_{ik} \left( \delta_{ki}(x_{k})^{2}+2x_{i} x_{k}\right) = a_{ik} \left( (x_{i})^{2}+2x_{i} x_{k}\right) \end{equation}