CESÀRO–VOLTERRA PATH INTEGRAL FORMULA ON A SURFACE

Abstract

If a symmetric matrix field e = (eij) of order three satisfies the Saint-Venant compatibility relations in a simply-connected open subset Ω of ℝ3, then e is the linearized strain tensor field of a displacement field v of Ω, i.e. [Formula: see text] in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is smooth, the unknown displacement field v(x) at any point x ∈ Ω can be explicitly written as a path integral inside Ω with endpoint x, and whose integrand is an explicit function of the functions eij and their derivatives. Now let ω be a simply-connected open subset in ℝ2 and let θ : ω → ℝ3 be a smooth immersion. If two symmetric matrix fields (γαβ) and (ραβ) of order two satisfy appropriate compatibility relations in ω, then (γαβ) and (ραβ) are the linearized change of metric and change of curvature tensor field corresponding to a displacement vector field η of the surface θ(ω). We show here that a "Cesàro–Volterra path integral formula on a surface" likewise holds when the fields (γαβ) and (ραβ) are smooth. This means that the displacement vector η(y) at any point θ(y), y ∈ ω, of the surface θ(ω) can be explicitly computed as a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γαβ and ραβ and their covariant derivatives. Such a formula has potential applications to the mathematical analysis and numerical simulation of linear "intrinsic" shell models.