Nonlocal vector calculus on the sphere

TL;DR

This paper develops a nonlocal vector calculus framework on the sphere using integral operators, proving a nonlocal Stokes theorem and showing convergence to classical calculus as interaction range diminishes.

Contribution

It introduces the first nonlocal vector calculus on curved surfaces, establishing a nonlocal Stokes theorem and demonstrating convergence to classical operators.

Findings

Operators are diagonalizable via spherical harmonics

Established a nonlocal Stokes theorem on the sphere

Proved strong convergence to classical vector calculus operators

Abstract

We introduce a nonlocal vector calculus on the unit two-sphere using weakly singular integral operators. Within this framework, the operators are diagonalizable in terms of scalar and vector spherical harmonics, a property that facilitates the proof of a nonlocal Stokes theorem. This constitutes the first instance of such a theorem on a curved surface. Furthermore, our analysis demonstrates the strong convergence of these nonlocal operators to the classical differential operators of vector calculus as the interaction range tends to zero.