Let
$\Omega $
be a volume occupied by a part of the body in the current
configuration, and
$\Gamma $
the boundary of the body. In the Lagrangian formulation,
$\Omega $
is the volume of space occupied by the material at the current
time, which is different from the Eulerian approach where a volume of space through which the
material passes is examined.
$\tau $
is the traction surface on
$\Gamma $
and $b$
are the body
forces.
Force equilibrium for the volume is then:
(1)
$$\underset{\Gamma}{\overset{}{\int}}{\tau}_{i}d\Gamma +{\displaystyle \underset{\Omega}{\int}\rho}}{b}_{i}d\Omega ={\displaystyle \underset{\Omega}{\int}\rho \frac{\partial {v}_{i}}{\partial t}d\Omega$$
Where,

$\rho $
 Material density
The Cauchy true stress matrix at a point of
$\Gamma $
is defined by:
(2)
$${\tau}_{i}={n}_{j}{\sigma}_{ji}$$
Where,
$n$
is the outward normal to
$\Gamma $
at that point. Using this definition,
Equation 1 is
written:
(3)
$$\underset{\Gamma}{\int}{n}_{j}{\sigma}_{ji}d\Gamma}+{\displaystyle \underset{\Omega}{\int}\rho}{b}_{i}d\Omega ={\displaystyle \underset{\Omega}{\int}\rho \frac{\partial {v}_{i}}{\partial t}d\Omega$$
Gauss' theorem allows the rewrite of the surface integral as a volume integral so
that:
(4)
$$\underset{\Gamma}{\int}{n}_{j}{\sigma}_{ji}d\Gamma}={\displaystyle \underset{\Omega}{\int}\frac{\partial {\sigma}_{ij}}{\partial {x}_{j}}d\Omega$$
As the volume is arbitrary, the expression can be applied at any point in the body providing the
differential equation of translation equilibrium:
(5)
$$\frac{\partial {\sigma}_{ij}}{\partial {x}_{j}}+\rho {b}_{i}=\rho \frac{\partial {v}_{i}}{\partial t}$$
Use of Gauss' theorem with this equation leads to the result that the true Cauchy stress matrix
must be symmetric:
(6)
$$\sigma ={\sigma}^{T}$$
So that at each point there are only six independent components of stress. As a result, moment
equilibrium equations are automatically satisfied, thus only the translational equations of
equilibrium need to be considered.