The discipline of ring spinning — the dominant technology for the production of staple-fibre yarns worldwide — is governed at every operational stage by the laws of classical mechanics and fluid dynamics. Yet the mathematical language that underlies these laws, namely differential and integral calculus, remains largely implicit in practitioner literature, scattered across specialised treatises in textile physics rather than presented as a unified, zone-resolved framework accessible to the engineering practitioner. This article addresses that gap by developing a systematic, zone-by-zone account of how the derivative operator and the definite integral appear in, and give precise mathematical form to, the five principal physical zones of the ring-spinning system: the drafting zone, the twist-insertion zone, the yarn balloon zone, the ring–traveller interaction zone, and the package winding and end-breakage zone. Within this theoretical framework, seventeen governing equations are derived or stated, each rigorously attributed to its physical origin and each accompanied by an explicit statement of the physical meaning of its mathematical operator. The drafting zone is analysed through the spatial derivative of fibre velocity and the mass-continuity equation, yielding a direct link between velocity gradients and yarn linear-density irregularity. Twist insertion is formalised as a definite integral of local twist rate along the yarn path, and the Euler–Eytelwein capstan equation is derived as the exponential solution of a first-order linear ODE. The yarn balloon is shown to be the solution surface of a second-order nonlinear ODE, with aerodynamic drag and tension profile recovered as definite integrals over the balloon height. Traveller tribodynamics are characterised through the second time-derivative of radial position and the time integral of frictional power dissipation. Finally, winding geometry is expressed as a polar arc-length integral, and yarn end-breakage probability is cast as the Weibull cumulative distribution function. This article presents a theoretical framework; no experimental measurements are reported herein. All figures are theoretical and illustrative; parameter values are selected for pedagogical clarity only, and all uncalibrated parameters carry explicit calibration disclaimers. The scope is confined to conventional ring spinning of staple-fibre yarns under steady-state operating assumptions and within the parameter values stated for each zone.