/* ---------------------------------- */
/* save as x_surfzx.h                 */
/* ---------------------------------- */
double IntSurf_dzdx(
double (*P_g)(double x, double y, double z),
double (*P_h)(double x, double z),
double  h,
p2d p   /* Here : p.x = p.x  */
)           /*        p.y = p.z  */
{
     return((      (*P_g)(p.x,(*P_h)(p.x,p.y),p.y)  *
                    sqrt( pow(fxy_x((*P_h),h,p),2) +
                          pow(fxy_y((*P_h),h,p),2) +
                          1)
             ));
}
/* ---------------------------------- */
double   sint_dzx(
double (*P_g)(double x, double y, double z),

double (*P_h)(double x, double z),

double (*P_u)(double x),
double (*P_v)(double x),
   int nz,
   
double x,

double h
)
{
p2d    p;

   int i = 0;
double m = 0.;
double M = 0.;


 for(i = 0; i <= nz; i++)
 {
       if(i ==0 || i== nz){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)))/nz;
        p.x = x;


  M += m * (      (*P_g)(p.x,(*P_h)(p.x,p.y),p.y)  *
                    sqrt( pow(fxy_x((*P_h),h,p),2) +
                          pow(fxy_y((*P_h),h,p),2) +
                          1));
 }

  return( ((((*P_v)(x)) -((*P_u)(x)))*M) / (3*nz) );
}
/* ---------------------------------- */
double surface_dzdx(
double (*P_g)(double x, double y, double z),

double (*P_h)(double x, double z),

double (*P_u)(double x),
double (*P_v)(double x),
   int nz,
   
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 *              sint_dzx( (*P_g),(*P_h),
                                  (*P_u),(*P_v),
                                    nz,                                 
                                  (ax + i*(bx-ax)/nx),
                                    h);
 }

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