Solution
= Solution
Differentiate $\lambda_s{}^c\mu_a{}^s=\delta_a{}^c$ to obtain
$$
\mu_a{}^s(T_b\lambda_s{}^c)
=-(T_b\mu_a{}^r)\lambda_r{}^c.
$$
Multiplying part viii by $\lambda_r{}^c(z)$ consequently gives
$$
\boxed{
\mu_a{}^s\mu_b{}^t
\frac{\partial^2z^r}{\partial y^s\partial y^t}\lambda_r{}^c(z)
=-[T_b(y)\mu_a{}^r(y)]\lambda_r{}^c(y)
+[T_b(z)\mu_a{}^r(z)]\lambda_r{}^c(z)}.
$$