What is the condition for $u dx + v dy + w dz$ to be exact differential?

Question

I think the condition is $curl(u,v,w) = 0$. But a book I am referring gives following condition:

$$u \Big( \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \Big) + v \Big( \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \Big) + w \Big( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \Big) = 0$$

Which one of the above conditions is right?