How to get the equality?

Question

How to get the equality 1,2,3 in picture below .$M$ is compact Riemannian manifold, i,e, $\partial\Omega=\varnothing$.

In fact ,I have a relaxed compute about there 3 equality.I just want see the standard version.

Answer 1

About the equality 1, $$ \int_M\nabla_i(e^{-f}\nabla_jv_{ij})dV=\int_M e^{-f}\nabla_i(\nabla_jv_{ij})dV-\int_Me^{-f}\nabla_if\nabla_jv_{ij}dV $$ then,let$F=\{e^{-f}\nabla_jv_{1j},...,e^{-f}\nabla_jv_{nj}\}$,so, $$ \int_M\nabla_i(e^{-f}\nabla_jv_{ij})dV=\int_M\nabla \cdot F dV=\int_{\partial M} F\cdot dS=0 $$ So, I can get the equality 1.

About the equality 2, $$ \int_M(\nabla_if\nabla_jv_{ij}-v_{ij}\nabla_if\nabla_jf)e^{-f}dV =\int_M\nabla_if(e^{-f}\nabla_jv_{ij}+v_{ij}\nabla_je^{-f})dV =\int_M\nabla_if(\nabla_j(e^{-f}v_{ij}))dV $$ then, $$ 0=\int_M\nabla_j(e^{-f}v_{ij}\nabla_if)dV =\int_m e^{-f}v_{ij}\nabla_i\nabla_jfdV +\int_M\nabla_if(\nabla_j(e^{-f}v_{ij}))dV $$ So, the equality 2 is right