Integral surface

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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.