Editing Method of characteristics (section)

===Fully nonlinear case===
Consider the partial differential equation

{{NumBlk|:|<math>F(x_1,\dots,x_n,u,p_1,\dots,p_n)=0</math>|{{EquationRef|4}}}}

where the variables ''p''<sub>i</sub> are shorthand for the partial derivatives

:<math>p_i = \frac{\partial u}{\partial x_i}.</math>

Let (''x''<sub>i</sub>(''s''),''u''(''s''),''p''<sub>i</sub>(''s'')) be a curve in '''R'''<sup>2n+1</sup>.  Suppose that ''u'' is any solution, and that

:<math>u(s) = u(x_1(s),\dots,x_n(s)).</math>

Along a solution, differentiating ({{EquationNote|4}}) with respect to ''s'' gives{{sfn|John|1991|pp=19-24}}

:<math>\sum_i(F_{x_i} + F_u p_i)\dot{x}_i + \sum_i F_{p_i}\dot{p}_i = 0</math>

:<math>\dot{u} - \sum_i p_i \dot{x}_i = 0</math>

:<math>\sum_i (\dot{x}_i dp_i - \dot{p}_i dx_i)= 0.</math>

The second equation follows from applying the [[chain rule]] to a solution ''u'', and the third follows by taking an [[exterior derivative]] of the relation <math>du - \sum_i p_i \, dx_i = 0</math>.  Manipulating these equations gives

:<math>\dot{x}_i=\lambda F_{p_i},\quad\dot{p}_i=-\lambda(F_{x_i}+F_up_i),\quad \dot{u}=\lambda\sum_i p_iF_{p_i}</math>

where λ is a constant.  Writing these equations more symmetrically, one obtains the Lagrange–Charpit equations for the characteristic

:<math>\frac{\dot{x}_i}{F_{p_i}}=-\frac{\dot{p}_i}{F_{x_i}+F_up_i}=\frac{\dot{u}}{\sum p_iF_{p_i}}.</math>

Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the [[Monge cone]] of the differential equation should everywhere be tangent to the graph of the solution.