The General Theory for One First-Order Equation (Continued)

Abstract

We are considering a general first-order partial differential equation F(x,y,p) = 0, where x = (x ₁,...,x _n), p = (p ₁,...,p _n), $p_i = \frac{{\partial u}} {{\partial x_i }},$ and y = u(x) is an unknown function. The equation determines a 2n-dimensional hypersurface V ²ⁿ in the space J ¹ of 1-jets of functions of (x ₁,... ,x _n). Each function has a 1-graph in J ¹; it is a solution of the equation if its 1-graph is a submanifold in V ²ⁿ.