Euler's second law for the conservation of angular momentum
The divergence theorem
The equilibrium equation (Cauchy's first law)
Let us first look at the first term of equation~(1) and apply the
divergence theorem (2).
Plugging this back into equation~(1) gives
Therefore, bringing all terms to the left hand side,
Using the equilibrium equations~(3), equation~(5) reduces
Since this holds for any , we have
If you work this expression out, you will see that .
Hence, the stress tensor is symmetric.