In the above calculation of $w(z)$, why would a term containing $\frac{\partial}{\partial z_j}$ appear? Since all of the substitutions in the formula don't have anything containing derivative's.
2026-03-25 17:36:24.1774460184
Poincaré-Dulac for vector field
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Namely, if a vector field $f=f(z)$ has components $f_j(z)$, then we write $$ f(z)=\sum_{j=1}^nf_j(z)\frac{\partial}{\partial z_j}. $$