Derivation of normal surface vector of a quasilinear PDE

[nigels]

Hi group,

In order to understand the methods of characteristics, I've been reading its wiki entry plus other sources. However, one of the first step of finding the normal surface vector given the PDE remains baffling to me in terms of how it's derived. In short, when provided with

$a(x, y, z) \frac{\partial{z}}{\partial{x}} + b(x, y, z) \frac{\partial{z}}{\partial{y}} = c(x,y,z)$

how does one derive the surface normal as

$\left( \frac{\partial{z}}{\partial{x}}(x,y), \frac{\partial{z}}{\partial{y}}(x,y), -1 \right)$

? What are the in-between steps? Thanks!

 

[hunt_mat]

The key important step is to understand for a surface defined as $\varphi (x,y,z)=0$, what the tangent vector is given by and what the normal vector is at a point. Once you know this then your geometrical understanding will give you the answer.