(10 pts.) The state of stress in a solid circular cylindrical shaft of radius R and length $l$ during torsion is given as follows
\begin{equation*}
T(x) = \begin{bmatrix}
0 & 0 & \mu \beta x_1 \\
0 & 0 & -\mu \beta x_2 \\
\mu \beta x_1 & -\mu \beta x_2 & 0
\end{bmatrix},
\end{equation*}
where $\mu$ is the shear modulus, $\beta = d\theta/dx_3$ is the rate of twist, both are constants, and $\theta$ is the angle of twist.
a) If the body force $\mathbf{b} = \mathbf{0}$, show that $T(x)$ satisfies local equilibrium.
b) Calculate the net traction force acting on each surface of the shaft.
