For shallow geometry (neglecting squares of first order partial derivatives) of a two parameter $(x,y) $ surface embedded in $\mathbb R^3.$ 

From the classical St. Venant relation using rectangular coordinates, we have the following scalar invariant associated with ***large non-linear deformations***:

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

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