/* ---------------------------------- */
/* save as x_fdydx.h                   */
/* ---------------------------------- */
double   fint_dyx(
double (*P_M)(double x, double y, double z),
double (*P_N)(double x, double y, double z),
double (*P_P)(double x, double y, double z),

double (*P_f)(double x, double y),
double (*P_u)(double y),
double (*P_v)(double y),
   int ny,

double x,

double h
)
{
v3d  n;
p2d   p;
   int i = 0;
double m = 0.;
double M = 0.;


 for(i = 0; i <= ny; i++)
 {
       if(i ==0 || i== ny){m = 1.;}
  else if(fmod(i,2) == 0){m = 2.;}
  else                   {m = 4.;}

        p.y = ((*P_u)(x)) + i*(((*P_v)(x))-((*P_u)(x)))/ny;
        p.x = x;

  n = n_xy((*P_f), p, h);	
  
  M += m * ( ( n.i*(*P_M)(p.x, p.y, (*P_f)(p.x,p.y)) +
           n.j*(*P_N)(p.x, p.y, (*P_f)(p.x,p.y)) +
           n.k*(*P_P)(p.x, p.y, (*P_f)(p.x,p.y)) ) *

          sqrt( pow(fxy_x((*P_f),h,p),2) +
                pow(fxy_y((*P_f),h,p),2) +
                                      1)
             );
 }

  return( ((((*P_v)(x)) -((*P_u)(x)))*M) / (3*ny) );
}
/* ---------------------------------- */
double flux_dydx(
double (*P_M)(double x, double y, double z),
double (*P_N)(double x, double y, double z),
double (*P_P)(double x, double y, double z),

double (*P_f)(double x, double y),
double (*P_u)(double y),
double (*P_v)(double y),
   int ny,

double ax,
double bx,
   int nx,

double h
)
{
   int i = 0;
double m = 0.;
double M = 0.;

 for(i = 0; i <= nx; i++)
 {
       if(i ==0 || i== nx){m = 1.;}
  else if(fmod(i,2) == 0){m = 2.;}
  else                   {m = 4.;}

  M += m * fint_dyx( (*P_M),(*P_N),(*P_P),
                     (*P_f), (*P_u),(*P_v),
                       ny,                     
                     (ax + i*(bx-ax)/nx),
                      h);
 }

  return( ((bx -ax)*M) / (3*nx) );
}
/* ---------------------------------- */
/* ---------------------------------- */