The balance of angular momentum in an inertial frame can be expressed as:
We assume that there are no surface couples on or body couples in Ω. Recall the general balance equation
In this case, the physical quantity to be conserved the angular momentum density, i.e., . The angular momentum source at the surface is then and the angular momentum source inside the body is . The angular momentum and moments are calculated with respect to a fixed origin. Hence we have
Assuming that Ω is a control volume, we have
Using the definition of a tensor product we can write
Also, . Therefore we have
Using the divergence theorem, we get
To convert the surface integral in the above equation into a volume integral, it is convenient to use index notation. Thus,
where represents the ith component of the vector. Using the divergence theorem
Differentiating,
Expressed in direct tensor notation,
where is the thirdorder permutation tensor. Therefore,
or,
The balance of angular momentum can then be written as
Since Ω is an arbitrary volume, we have
or,
Using the identity,
we get
The second term on the right can be further simplified using index notation as follows.
Therefore we can write
The balance of angular momentum then takes the form
or,
or,
The material time derivative of is defined as
Therefore,
Also, from the conservation of linear momentum
Hence,
The material time derivative of ρ is defined as
Hence,
From the balance of mass
Therefore,
In index notation,
Expanding out, we get
Hence,
