Consider a straight bar $B$ of uniform cross-section whose axis is parallel to the $z$-axis of an $x y z$-coordinate system. Let $\Omega$ denote a typical cross-section of $B$ and assume the boundary $\partial \Omega$ is described by a smooth curve $C$ in the $x y$-plane as illustrated in the figure below.
(a) Let $\boldsymbol{r}(s)=x(s) e_{1}+y(s) \boldsymbol{e}_{2}, 0 \leq s \leq L$, be an arclength parametrization of $C$, so that
$$
\gamma=\frac{d \boldsymbol{r}}{d s}=\frac{d x}{d s} \boldsymbol{e}_{1}+\frac{d y}{d s} \boldsymbol{e}_{2}
$$
is a unit tangent vector field on $C$ in the direction of increasing arclength parameter $s$. Show that the vector field
$$
\boldsymbol{n}=\frac{d y}{d s} \boldsymbol{i}-\frac{d x}{d s} \boldsymbol{j}
$$
is a unit vector field on $C$, is everywhere orthogonal to $\gamma$, and is oriented such that $\boldsymbol{n} \times \gamma=\boldsymbol{e}_{3}$.
(b) Suppose the ends of the bar $B$ are twisted relative to each other by an amount so small that the configuration of $B$ remains essentially unchanged; in particular, cross-sections do not warp and remain perpendicular to the $z$-axis. For such twisting, we may assume that the Cauchy stress in $B$ is of the form
$$
[\boldsymbol{S}]=\left(\begin{array}{ccc}
0 & 0 & \tau_{x} \\
0 & 0 & \tau_{y} \\
\tau_{x} & \tau_{y} & 0
\end{array}\right)
$$
where $\tau_{x}$ and $\tau_{y}$ are each functions of $x$ and $y$ only, and that these two functions are related to a single scalar function $\phi(x, y)$ by
$$
\tau_{x}=\frac{\partial \phi}{\partial y}, \quad \tau_{y}=-\frac{\partial \phi}{\partial x}
$$
Determine the boundary condition imposed on the function $\phi$ by the requirement that $\boldsymbol{S} \boldsymbol{n}=\mathbf{0}$ on $C$. That is, what conditions must $\phi$ satisfy on $C$ in order that the lateral surface of the bar be traction-free?