Landau and Lifshitz argument for symmetry of stress tensor

In Landau and Lifshitz's book on the theory of elasticity (vol 7, theoretical physics series), specifically section 2 of the first chapter, the authors present an argument for justifying the symmetry of the stress tensor. On page 6, they derive an expression for the net moment of the force acting on a portion of a body due to the other parts of the body.

$$M_{ik}=\int\left(F_ix_k-F_kx_i \right) dV$$

They write, "Like the total force on any volume, this moment can be expressed as an integral over the surface bounding the volume."

What is the logic behind this statement? I understand the net force being solely a surface integral since the molecular forces resulting from a deformation are essentially short-range. They go on to use this to prove that the stress tensor is symmetric.

Note: The only assumption declared prior to this argument is that the aforementioned internal forces are short-range. No other assumptions seem to have been declared. (Please correct me if I'm wrong)

• Commented Jul 24, 2018 at 21:04
• Landau routinely assumed that you knew what the assumptions were. Don't get me wrong, I love the L&L books, but while they may be at what Landau considered an introductory level, mere mortals should fear to tread without other resources to consult. Commented Jul 24, 2018 at 22:54

One way to interpret it is that they're assuming you can apply the divergence theorem backwards; that is, that the integrand $\left(F_ix_k-F_kx_i \right)$ is itself the divergence of some well-defined vector field $\mathbf{\tau}$, which is connected to the traction vector field $\mathbf{T}$.