Integral surface

For example, consider the quasilinear PDE $a(x,y,u)u_{x}+b(x,y,u)u_{y}=c(x,y,u)\,$ and the graph $z=u(x,y)\,$.
Let $z_{0}=u(x_{0},y_{0})\,$.
At the point $(x_{0},y_{0},z_{0})\,$, the normal vector is $N_{0}=<-u_{x}(x_{0},y_{0}),-u_{y}(x_{0},y_{0}),1>\,$.
The PDE implies that the vector $V_{0}=\,$ is perpindicular to the normal vector and therefore must lie in the tangent plane to the graph of $z=u(x,y)\,$ at the point $z_{0}\,$.
Now $V(x,y,z)=\,$ defines a vector field in ${\mathbb {R}}^{3}\,$ to which graphs of solutions must be tangent at each point. Surfaces that are tangent at each point to a vector field in ${\mathbb {R}}^{3}\,$ are called integral surfaces of the vector field. Curves tangent at each point are called integral curves.