Integral surface

From Exampleproblems

In mathematics, an integral surface is a smooth union of characteristic curves. The surface is tangent to each point in a given vector field.

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 $V_0=<a(x_0,y_0,z_0),b(x_0,y_0,z_0),c(x_0,y_0,z_0)>\,$ 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 $V(x,y,z)=<a(x,y,z),b(x,y,z),c(x,y,z)>\,$ 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.