Re: euler without d/dt? time-independent conservation law.

let's rewrite your equation to
(*) fp(u)*u_x + gp(u)*u_y=0 where fp(u) is the derivate of f at
u(x,y).

Now make the ansatz (characteristic) y= y(x), then

(**) d/dx (u(x,y(x))= u_x + u_y*yp where yp= d/dx y .
Assume we can divide (*) by fp(u), then we get by plugging (*)
into (**) :

d/dx (u(x,y(x))= (-gp(u)/fp(u) + yp) * u_y = (**)

Now, if we solve the ODE yp(x)= gp(u(x,y(x))) / fp(u(x,y(x)),
then (***) = 0, or u(x,y(x)) = constant.

I.e., along the so-called characteristic curve (x,y(x))
the solution u is constant,

Thanks very much for taking the time to comment.
The method of characteristics works well for a system of 2 PDEs and 2
unknown functions where it is easy to identify the so called Riemann
Invariants but what if you have 4 of each? This is this issue that
struggling with here.


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