In the second edition of Lee's book of smooth manifolds (pg 510), there is a passage that i do not understand.
Proposition $(19.28)$: Suppose $\alpha$ and $\beta$ are smooth real-valued functions definined on some open subset $W \subset \mathbb{R}^{3}$ and satistying $(19.14)$ there. For each $(x_{0},y_{0},z_{0})$, there exist a neighborhood $U$ of $(x_{0},y_{0})$ em $\mathbb{R}^{2}$ and a unique function $u:U\to \mathbb{R}$ satisfying $(19.13)$ and $u(x_{0},y_{0})=z_{0}$.
A passage is in the page $511$: "..Let $\Phi:V \to \mathbb{R}$ be a defining function for $N$ on some neighborhood $V$ of $p$; for example we could take $\Phi$ to be the third coordinate function in a flat chart. .."
$(19.13)$
- $\frac{\partial u}{\partial x}(x,y)= \alpha(x,y,u(x,y))$
- $\frac{\partial u}{\partial y }(x,y)= \beta(x,y,u(x,y))$
$(19.14)$ $\frac{\partial \alpha}{\partial y}+\beta\frac{\partial \alpha}{\partial z}= \frac{\partial \beta}{\partial y} + \alpha \frac{\partial \beta}{\partial z}$
Any answers are appreciated.