Just not sure if I went about this calculation the correct way (note $k_{ij}$ is symmetric) \begin{equation} \frac{1}{2}k_{ij}\frac{\partial }{\partial q^k} (e^{q^i+q^j}) \end{equation}
- $\frac{\partial q^i }{\partial q^j}=\delta_{ij}$
- By applying what was just stated one gets $e^{q^i+q^j}(\delta_{ik}+\delta_{jk})$
The issue arises when I continue from here . The reason I use only allows me to come to a solution if I expand the previous result
- = $e^{q^i+q^j}\delta_{ik} +e^{q^i+q^j}\delta_{jk}$ (1)
In the first additive term i->k and in the second j->k (didnt include $k_{ij}$ for for succinctness)
which gives
\begin{equation} \frac{1}{2}k_{ij}\frac{\partial }{\partial q^k} (e^{q^i+q^j})= \frac{1}{2}(k_{kj}e^{q^k+q^j}+k_{ik}e^{q^i+q^k}) \end{equation}
Now just sum just changed dummy index's within the two additive terms and use the fact that $k_{ij}$ is symmetric to get \begin{equation} =\frac{1}{2}(k_{kj}e^{q^k+q^j}+k_{kj}e^{q^j+q^k}) = k_{ij}e^{q^i+q^j} \end{equation}
Just changed the free index as I changed it through the whole expression
So is there away I can go about this without having to expand it in (1) and is this even correct ?
Any recommendations for reference books on index notion and operations with them would be much appropriated