Substitution in partial differential equation

Question

The setting is the following: we have $\frac{\partial^2\phi}{\partial^2 y} - \frac{\partial^2\phi}{\partial^2 x} = 0$ and if we reduce this to a first order pde by substituting $u = \frac{\partial\phi}{\partial x} \text{, } v = \frac{\partial\phi}{\partial y}\text{ and } \frac{\partial}{\partial x} = \xi \text{, } \frac{\partial}{\partial y} = \zeta$, we get the following equations: $\xi \dot{v} + \zeta \dot{u} = 0 \text{ and } -\xi \dot{u} + \zeta \dot{v} = 0$. My question is why is there a $\dot{u}$ and a $\dot{v}$ after the substitution? Should it not be just $u$ and $v$ without a dot? Thank you.