I am reading *Harmonic Mappings of Riemannian Manifolds* by Eells and Sampson. In chapter 2, the author(s) used the divergence theorem, which does not look like the usual divergence theorem for manifolds, and I can't figure out why it is correct.

Simply speaking, in this paper, a family of quantities $\xi^j\ (j=1,\cdots,m)$ is defined using local coordinates $(x^1,\cdots,x^m)$ on a Riemannian manifold $M$. And one can check $\xi^j$ vary contravariantly under coordinate changes, and therefore define a tangent vector field on $M$ (Einstein's summation convention used): $$X=\xi^j\frac{\partial}{\partial x^j}$$ Then the paper says and I quote:

...whence by the Green's divergence theorem $$\int_M\xi^j_jdM=0.$$

where $dM$ is the volume element on $M$.

There are two things I am not sure of.

- I am not sure what $\xi^j_j$ here means. My best guess would be $\xi^j_j=\frac{\partial\xi^j}{\partial x^j}$.
- Suppose $\xi^j_j$ does mean as above. What I don't get is why $\int_M\xi^j_jdM=0$ follows from the divergence theorem. According to Proposition 2.46 of Lee's book
*Introduction to Riemannian Manifolds*, the coordinate representation of the divergence of $X=\xi^j\frac{\partial}{\partial x^j}$ should be ($\det g$ is the determinant of the matrix for the metric tensor under local coordinates) $${\rm div} X=\frac{1}{\sqrt{\det g}}\frac{\partial}{\partial x^j}(\xi^j\sqrt{\det g})$$ The divergence theorem then takes the form ($M$ is assumed to be compact without boundary so that the RHS vanish): $$\int_M(\xi^j_j+\frac{\xi^j}{\sqrt{\det g}}\frac{\partial\sqrt{\det g}}{\partial x^j})dM=0$$ There's clearly an extra term.

Any help is appreciated.