Stokes’ Theorem for Spiral Paths

Abstract

The macroscopic flow along the boundary of a closed curve is equivalent to the cumulative sum of microscopic flows within the enclosed area. Green’s theorem formalizes this relationship by connecting the counterclockwise flow within the surface of a two-dimensional manifold to the counterclockwise flow along its boundary. Building on this, Stokes’ theorem (henceforth ST) extends the concept to three-dimensional manifolds. By converting a surface integral of the curl of a vector field over the surface into a line integral around the boundary, ST enables the assessment of surface flows based on their boundary flows. In this context, we propose a further generalization of ST to include helicoidal spiral paths. This extension is applicable to a wide range of physical and biological systems where spiral motion plays a significant role, providing a robust framework for in-depth analysis of complex dynamical systems across multiple disciplines.