# Integral surface

From Example Problems

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 and the graph .

Let .

At the point , the normal vector is .

The PDE implies that the vector is perpindicular to the normal vector and therefore must lie in the tangent plane to the graph of at the point .

Now defines a vector field in to which graphs of solutions must be tangent at each point. Surfaces that are tangent at each point to a vector field in are called integral surfaces of the vector field. Curves tangent at each point are called integral curves.