The balance of linear momentum can be expressed as:
where is the mass density, is the velocity, is the Cauchy stress, and is the body force density.
Recall the general equation for the balance of a physical quantity
In this case the physical quantity of interest is the momentum density, i.e., . The source of momentum flux at the surface is the surface traction, i.e., . The source of momentum inside the body is the body force, i.e., . Therefore, we have
The surface tractions are related to the Cauchy stress by
Therefore,
Let us assume that Ω is an arbitrary fixed control volume. Then,
Now, from the definition of the tensor product we have (for all vectors )
Therefore,
Using the divergence theorem
we have
or,
Since Ω is arbitrary, we have
Using the identity
we get
or,
Using the identity
we get
From the definition
we have
Hence,
or,
The material time derivative of ρ is defined as
Therefore,
From the balance of mass, we have
Therefore,
The material time derivative of is defined as
Hence,
