Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates:
$$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} }, t=\frac{\partial ^2w}{{\partial {y} ^2 } },s=\frac{\partial ^2w}{{\partial {x} \partial {y} } };\quad K= \frac{rt-s^2}{(1+p^2+q^2)^2}; $$
For shallow shell geometry we neglect squares of first order partial derivatives $ (p^2,q^2) $ in its denominator for surface embedded in $\mathbb R^3.$ The standard differential relation canbe 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:
$$ \dfrac{\partial ^2{\epsilon_x}}{{\partial {y^2}} }+ \dfrac{\partial ^2{\epsilon_y}}{{\partial {x^2}} }- \dfrac{\partial ^2{\gamma_{xy}}} {\partial {x}\partial {y} } $$
It links direct and shear strains for compatibility and is known to represent Gauss curvature as was derived by Von Kármán. It is an isometric invariant that can be derived from first fundamental form of surface theory. For example please see reference Chap 13, page 417 Equation (c) Theory of Plates and Shells by S, Timishenko
dealing with large deformations in both rectangular and polar coordinates.
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?
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