For shallow geometry ( neglecting squares of first order patial derivatives) of an $\mathbb R^2 $ surface imbedded $\mathbb R^3 $ we have from classical St. Venant relation using rectangular coordinates a scalar invariant:

$$  \dfrac{\partial ^2{\epsilon_x}}{{\partial {y^2}} }+ \dfrac{\partial ^2{\epsilon_y}}{{\partial {x^2}} }- \dfrac{\partial ^2{\gamma_{xy}}} {\partial {x}\partial {y} }  $$

It links direct and shear strains for compatibility and is known to represent Gauss curvature derived by Von Karman.  It is an isometric invariant that can derived from first fundamental form of surface theory.

But what about the other geometrical invant of St. Venant,

$$  \dfrac{\partial }{{\partial {x}}}  \big[ \dfrac{\partial {\gamma_{zx}}} {\partial {y} }+  \dfrac{\partial {\gamma_{xy}}} {\partial {z} } -\dfrac{\partial {\gamma_{yz}}} {\partial {x} } \big] -  2\dfrac{\partial ^2{\epsilon_x}}{{\partial {y} \partial {z}  } } ?$$

Is it  composed of isometric/ topological invariants derived from first and second fundamental forms? Or the Gauss-Bonnet theorem? How is this derived and recognized?

Regards