Problems with indices and chain rule

Question

Is there way to handle $\sum_i\frac{\partial}{\partial p_j}p_i\ln p_i$ with the Einstein summation convention without getting troubles with indices?

$\frac{\partial}{\partial p_j}p_i\ln p_i=\delta_{ij}\ln p_i+\frac{p_i}{p_i}\delta_{ij}$

Answer 1

The Einstein summation convention is that any index which is repeated implies a summation over that index. For example $$\eqalign{ p_i(\log p_i) \:&\equiv\: \sum_i p_i(\log p_i) \\ }$$ Extending the Kronecker delta symbol to three indexes $$\eqalign{ \def\LR#1{\left(#1\right)} \def\d{\vec\delta} \def\o{{\tt1}} \def\p{\partial} \def\qiq{\quad\implies\quad} \def\grad#1#2{\frac{\p #1}{\p #2}} \def\gradLR#1#2{\LR{\grad{#1}{#2}}} \d_{ijk} &= \begin{cases} 1 \quad&{\rm if\;} i=j=k \\ 0 \quad&{\rm otherwise} \\ \end{cases} \\ }$$ allows you to write the elementwise/Hadamard product of vectors using Einstein notation $$\eqalign{ c = a\odot b \qiq c_j = a_i\,\d_{ijk}\,b_k \\ }$$ If $q$ is the Hadamard inverse of $p,\,$ then it satisfies $$\eqalign{ \o = p\odot q \qiq \o_j = p_i\,\d_{ijk}\,q_k \\ }$$ and it can be used to write the elementwise derivative of $\log(p)$ in index notation $$\eqalign{ \p_j(\log p_i) &= \d_{jik}\,q_k \\ }$$ Putting it all together $$\eqalign{ \p_j\LR{p_i\log(p_i)} &= \LR{\p_jp_i}\log(p_i) + p_i \p_j\log(p_i) \\ &= \delta_{ij}\log(p_i) + p_i \d_{jik}\,q_k \\ &= \log(p_j) + \o_j \\ }$$

Answer 2

$\frac{\partial}{\partial p_j}(p_i\ln p_i)=\delta_{ij}(\ln p_i+\frac{p_i}{p_j})=0$ for all $i\neq j$, so we only need to consider the $j$-th term, which is $\ln p_j+1$.