I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation:
Let $\displaystyle \xi=\sum_{i,j}a_{i,j}(z)d\bar{z_i}\otimes\frac{\partial}{\partial z_j}.$ Now If we apply this "operator" to $\frac{\partial}{\partial \bar z_i}$, according to the author we get $\displaystyle\frac{\partial}{\partial \bar z_i} +\sum_{i}a_{i,j}(z)\frac{\partial}{\partial z_j}.$ But according to my calculations I should get only $\displaystyle\sum_{i}a_{i,j}(z)\frac{\partial}{\partial z_j}.$
Am I missing something? or is this a typo?
Any help is really appreciated.