A method of shell theory in determination of the surface from components of its two fundamental forms

We introduce the tensor which maps the Cartesian plane into the tangent plane of the surface. Then by analogy to the polar decomposition theorem widely used in the non‐linear theory of thin shells the tensor is represented as composition of the surface stretch and 3D rotation fields. Left and right polar decompositions are analysed. For each of them the position vector of the surface in space is established uniquely from the surface metric and curvature components in three subsequent steps: 1) the stretch field is found by pure algebra, 2) the rotation field is obtained by solving the system of linear first‐order PDEs, and 3) the position vector of the surface follows by quadrature. Integrability conditions for the rotation field are shown to be alternative forms of the Gauss‐Mainardi‐Codazzi equations. The results are illustrated by a simple analytically solved example. The proposed method is expected to be more appealing and in some cases also more efficient than those used in classical differential geometry. We also briefly discuss the relation of our method to the one associated with integrable surfaces and soliton equations.