I'm having a bit of trouble understanding how my book derived one of its equations. The example goes:
Consider the surfaces of the form $F(u,v)=0$ where $u=u(x,y,z)$ and $v=v(x,y,z)$ are known functions of $x, y$ and $z$, and $F$ is an arbitrary function of $u$ and $v$ having first order partial derivatives with respect to $u$ and $v.$
On differentiating $F(u,v)=0$ with respect to $x$ and $y$, treating $z$ as a function of $x$ and $y$, we get respectively
$$\frac{\partial F}{\partial u}(u_x + pu_z)+\frac{\partial F}{\partial v}(v_x + pv_z)=0,$$
$$\frac{\partial F}{\partial u}(u_y + qu_z)+\frac{\partial F}{\partial v}(v_y + qv_z)=0.$$
My question is where did $(u_x + pu_z)$, $(v_x + pv_z)$, $(u_y + qu_z)$ and $(v_y + qv_z)$ come from. I'm guessing they stand for $\frac{\partial u}{\partial x}$, $\frac{\partial v}{\partial x}$, $\frac{\partial u}{\partial y}$ and $\frac{\partial v}{\partial y}$ respectively, but I don't see how that's true. Can someone clarify?