Is it possible to further simplify the following equation?

Question

Is it possible to write the following equation in an even simpler form? (In other words does this have any specific implications on the form $\vec f(\vec x)$ can take?) $${\partial f_j(\vec x)\over \partial x_i}={\partial f_i(\vec x)\over \partial x_j}$$ where $x_k$ are components of $\vec x$ and $f_k(\vec x)$ are the components of $\vec f(\vec x)$.