studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor
$$S=\left(\frac{\pi + \mu_0}{\lambda_1} + \mu_0 \lambda_1\!\right)e_r \otimes e_1 + \left(\frac{\pi + \mu_0}{\lambda_2} + \mu_0 \lambda_2\!\right) e_\theta \otimes e_2$$
where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$
How is the divergence defined in this case? The book writes the following
$$Div(S) = \frac{\partial }{\partial x_1^0} S_{r1} e_r + \frac{\partial}{\partial x_2^0} S_{\theta 2} e_{\theta} = 0$$
but I can't see why this is the divergence of $S$. I'd like to have a formal proof of this. I know that, by definition, the divergence of a tensor field $S$ is the unique vector field such that
$Div(S^T a)=Div(S) \cdot a$
for every vector $a$