Solution
= Solution
Substitute
$$
\mathbf f=-\boldsymbol\gamma\mathbin{\cdot}\mathbf r_t
+\mathbf f^A
$$
into force balance. With $\boldsymbol\mu=\boldsymbol\gamma^{-1}$,
$$
\mathbf r_t
=\boldsymbol\mu\mathbin{\cdot}
(\partial_s\mathbf F+\mathbf f^A).
$$
Projection onto the local frame gives
$$
\boxed{
W=\mathbf t\mathbin{\cdot}\boldsymbol\mu
(\partial_s\mathbf F+\mathbf f^A),
\qquad
U=\mathbf n\mathbin{\cdot}\boldsymbol\mu
(\partial_s\mathbf F+\mathbf f^A)
}.
$$