/* ---------------------------------- */
/* save as x_fdxdy                   */
/* ---------------------------------- */
double   fint_dxy(
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 nx,
   
double y,

double h
)
{
  p2d p;
  
   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.;}

        p.x = ((*P_u)(y)) + i*(((*P_v)(y))-((*P_u)(y)))/nx;
        p.y = y;
  
  M += m * (
             -fxy_x((*P_f),h,p)*(*P_M)(p.x,p.y,(*P_f)(p.x,p.y))+ 
             -fxy_y((*P_f),h,p)*(*P_N)(p.x,p.y,(*P_f)(p.x,p.y))+ 
                              1*(*P_P)(p.x,p.y,(*P_f)(p.x,p.y))
            );
          
 }

  return( ((((*P_v)(y)) -((*P_u)(y)))*M) / (3*nx) );
}
/* ---------------------------------- */
double flux_dxdy(
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 nx,
   
double ay,
double by,
   int ny,
   
double h

)
{
   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.;}

  M += m *              fint_dxy( (*P_M),(*P_N),(*P_P),
                                  (*P_f), (*P_u),(*P_v),
                                    nx,                                  
                                   (ay + i*(by-ay)/ny),
                                    h);
 }

  return( ((by -ay)*M) / (3*ny) );
}
/* ---------------------------------- */
/* ---------------------------------- */