For shallow geometry (neglecting squares of first order partial derivatives) of an $\mathbb R^2 $ surface embedded $\mathbb R^3,$ from the classical St. Venant relation using rectangular coordinates, we have the following 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 Kármán. It is an isometric invariant that can be derived from first fundamental form of surface theory.

But what about the other geometric invariant 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 or topological invariants that are derived from first and second fundamental forms? Or the Gauss-Bonnet theorem? How is this derived and recognized?

Regards