Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates:

$$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} }, t=\frac{\partial ^2w}{{\partial {y} ^2 } },s=\frac{\partial ^2w}{{\partial {x} \partial {y} } };\quad K= \frac{rt-s^2}{(1+p^2+q^2)^2}; $$

where for a surface embedded in $\mathbb R^3 $ for shallow shell geometry we took $z=f(x,y)=w $ from Mongé form and neglected squares of first order partial derivatives $ (p ,q) $ in its denominator.The standard differential relation can be linked to classical large deformation theory of deformations of Plates and Shells.

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

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

It links direct and shear strains for compatibility and is known to represent Gauss curvature as was derived by Von Kármán. The full non-linear $K$ is a known isometric invariant that can be derived from Christoffel symbols of the first fundamental form of surface theory. Please refer to Chap 13, page 417 Equation (c) Theory of Plates and Shells text-book by S.Timishenko

Von Kármán Reln in Mechanics

dealing with large deformations in both rectangular and polar coordinates.

Saint Venant tensor $ W_{ijkl} $ vanishing for a simply connected domain implies that strain is a symmetric derivative of some vector field:

St. Venant Compatibility Wiki

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