Here we generalize our basic definitions of angular momentum and torque and study the resulting form of the balance of angular momentum equation. In particular, suppose the angular
momentum (about origin) of an arbitrary open subset $\Omega_{t}$ is given by
$$
j\left[\Omega_{t}\right]_{o}=\int_{\Omega_{t}} \rho \eta+\boldsymbol{x} \times \rho \boldsymbol{v} d V_{\boldsymbol{x}}
$$
where $\boldsymbol{\eta}(\boldsymbol{x}, t)$ is an intrinsic angular momentum field per unit mass, and suppose the resultant torque (about origin) of external influences on $\Omega_{t}$ is given by
$$
\tau\left[\Omega_{t}\right]_{o}=\int_{\Omega_{t}} \rho h+\boldsymbol{x} \times \rho \boldsymbol{b} d V_{x}+\int_{\partial \Omega_{t}} \boldsymbol{m}+\boldsymbol{x} \times \boldsymbol{t} d A_{\boldsymbol{x}}
$$
where $\boldsymbol{h}(\boldsymbol{x}, t)$ is an intrinsic body torque field per unit mass and $m(x, t)$ is an intrinsic surface torque field per unit area. Notice that the usual definitions of angular momentum and torque are recovered in the case when $\boldsymbol{\eta}, \boldsymbol{h}$ and $\boldsymbol{m}$ all vanish. In direct analogy to the relation $t=S n$ for surface traction, we assume $m=M n$ where $M(\boldsymbol{x}, t)$ is a second-order tensor field analogous to the Cauchy stress $\boldsymbol{S}(\boldsymbol{x}, t)$.
(a) Show that the local Eulerian form of the law of angular. momentum in Axiom $5.2$ is
$$
\rho \dot{\eta}=\nabla^{x} \cdot \boldsymbol{M}+\boldsymbol{\xi}+\rho \boldsymbol{h}, \quad \forall x \in B_{t}, t \geq 0
$$
where $\boldsymbol{\xi}(\boldsymbol{x}, t)=\epsilon_{i j k} S_{k j}(\boldsymbol{x}, t) \boldsymbol{e}_{i} .$ (Hint: The linear momentum equation is unchanged.)
(b) Assuming $M$ and $\boldsymbol{h}$ are identically zero, show that $\dot{\boldsymbol{\eta}}(\boldsymbol{x}, t)=$ $\mathbf{0}$ if and only if $\boldsymbol{S}(\boldsymbol{x}, t)$ is symmetric.