Hi can anyone explain the Liebnitz rule in the case of multi-index notation (i.e partial derivatives). How can we use the choose function on multi-indexes.
Wiki gives it as $D^{\alpha}(fg)=\sum_{\{\beta:\beta<\alpha\}}\binom{\alpha}{\beta}D^{\alpha-\beta}(f)D^{\beta}(g) $
Just as the differential operator $D^\alpha:=\prod_i\partial_i^{\alpha_i}$ and $\beta\le\alpha$ (you wouldn't use a strict inequality in your summation condition, by the way) means that $\beta_i\le\alpha_i$ for all $i$, binomial coefficients are defined $\binom{\alpha}{\beta}:=\prod_i\binom{\alpha_i}{\beta_i}$. An easy way to see that the formula isn't instead $\binom{|\alpha|}{|\beta|}$ is thay we require the coefficient vanish if $\beta_i>\alpha_i$ for some $i$, so that $\binom{\alpha_i}{\beta_i}=0$.