We consider any enclosed volume (&tau) capturing a fluid element.
We equate the added momentum throughout &tau over a perid of time dT,
with the impact of all the forces of &tau over the same period. This is
the equation of motion.
M_{&tau }(T+dT)  M_{&tau }(T) = &int (Du/Dt) &rho d &tau
where u is the velocity of the fluid element, and &rho is its specific
gravity, M  is momentum.
The forces that apply are either distance forces (external):
&int F_{external } &rho d &tau
where F is a force per unit mass;
or surface forces.
Surface forces apply to the entire surface (S) of &tau, and at each
surface area, the force is may be of any direction.
Accoutning for such direction variety of forces over the direction
variety of the surface elements of &tau is a complicated task.
It is handled via a dedicated mathematical construct: the tensor.
Tensor notation allows for identifying the impact of each force of any
direction to whatever the direction of the surface element it is
associated with. The overall impact is given by:
&int_{S } &sigma_{ij } dS_{j }
where i,j represent all possible directions for ds and for surface
forces  given a cartesian representation.
Applying the surface integral to volume integral identity (Gauss
theorem) we get:
&int_{S } &sigma_{ij } dS_{j } = &int_{&tau } (&part &sigma_{ij }/&part x_{j })d&tau
which allows us to express all integrals with respect to &tau, and hence:
&int _{&tau } &rho (Du_{i }/dt)d &tau = &int _{&tau } &rhoF_{i } d&tau + &int _{&tau } (&part &sigma_{ij }/&part x_{j })d&tau
which must be true for any arbitrary &tau, and hence;
&rho (Du_{i }/dt)d &tau = &rhoF_{i } d&tau + (&part &sigma_{ij }/&part x_{j })d&tau
which is the equation of motion.
