// nabla.h  del^2 = nabla = Laplacian  n dimensional space of function U(r,a)
//        n-dimensional sphere, radius r
//        a[0],...,a[n-2] are angles in radians (User can rename in their code)
//        compute Laplacian of U(r,a[]) at given angles
double nabla(int n, double r, double a[], double (*U)());

// alternate call if partial derivative values of U are known
//        da[0],...,da[n-2]   are first partial derivatives of users U(r,a)
//        dda[0],...,dda[n-2] are second partial derivatives of users U(r,a)
//                            with respect to a[0],...,a[d-2] evaluated
//                            at a[0],...,a[d-1]
double nablapd(int n, double r, double a[],
               double dr, double ddr, double da[], double dda[]);

// utility function for n-dimensional cartesian to spherical coordinates
// n>3 x[0..n-1]  r, a[0..n-2]
void toSpherical(int n, double x[], double *r, double a[]);

// utility function for n-dimensional spherical to cartesian coordinates 
// n>3 r, a[0..n-2] x[0..n-1]
void toCartesian(int n, double r, double a[], double x[]);

// utility function for n-dimensional U(r, a) at r, a[] 
// to derivatives da[] and dda[]
void sphereDeriv(int n, double (*U)(), double r, double a[],
                 double *dr, double *ddr, double da[], double dda[]);