Integral formulas for the r-mean curvature linearized operator of a hypersurface

For a normal variation of a hypersurface M-n in a space form Q(c)(n+1) by a normal vector field fN, R. Reilly proved: d/dt Sr+1(t)\(t=0)=L(r)f + (S1Sr+1-(r+2)Sr+2)f + c(n-r)S(r)f, where L-r (0 less than or equal to r less than or equal to n-1) is the linearized operator of the (r + 1)-mean curvature Sr+1 of M-n given by L-r = div(P-r del); that is, L-r = the divergence of the rth Newton transformation P-r of the second fundamental form applied to the gradient del, and L-0 = Delta the Laplacian of Mn. From the Dirichlet integral formula for L-r, integral(Mn) (f L(r)g+[P-r del f, del g]) = 0, new integral formulas are obtained by making different choices of f and g, generalizing known formulas for the Laplacian. The method gives a systematic process for proofs and a unified treatment for some Minkowski type formulas, via L-r.