Suppose we have
$$ a u_x + b u_y = c $$
Im given the following definition: ${\bf integral \; surface}$ for the pde above is the set
$$ \Sigma = \{(x,y,z) \in U \subset \mathbb{R}^3 : z = u(x,y) \} $$
where $U$ is open.
Question: In my notes, I read the following sentence:
Suppose $z = u(x,y)$ is an integral surface of the vector field V=(a,b,c).
What do they mean by that? I see in the definition of integral surface to be the solution of the pde, but what do they mean by integral surface of the vector field?
If $z = u(x,y)$ is a solution to $$ a u_x + b u_y = c \ \ \ (*), $$ then $z$ is called the integral surface of $(*)$. Moreover, the vector (field) $V = (a,b,c)$ is called the characteristic direction of $(*)$. We see why after simple rearrangement of $(*),$ which can be expressed as $$ \begin{pmatrix} u_x \\ u_y \\ -1 \end{pmatrix} \cdot \begin{pmatrix} a \\ b \\ c \end{pmatrix} = 0. $$ $(*)$ in this form has a more visible geometric meaning. The normal direction of $u$ at each point is orthogonal to $V$. That is, $V$ is, at every point, tangent to the surface $u$.