Notation of a vector field

Question

Usually vector field looks like $\sum_{i} a_i(x_1,x_2,...x_n) \frac{\partial}{\partial x_i}$, also it is not a problem to write it like that $\sum_{i}a_i\partial_i$ or even $a^i\partial_i$ using Einstein notation. However i cannot understand a formulation of a problem, where we are to consider all such vector fields that satisfy the equation: $\partial_iu_j+\partial_ju_i=0$, where $u_i$ are the standard coordinates on $\mathbb{R}^n$ (original words from the problem sheet). What does that mean?