The Role and Application of Calculus in Aerodynamics: A Systematic Review

Abstract

This review brings together 18 studies that explore how calculus is applied in aerodynamics, drawing from leading aerospace and engineering databases. Among these, seven studies (39%) concentrated on differential equations especially the Navier–Stokes and Euler formulations which remain the core tools for representing fluid flow and turbulence. Another five studies (28%) examined the use of the calculus of variations and shape calculus in optimization tasks, including wing design and trajectory planning, and consistently reported gains in aerodynamic efficiency. Four studies (22%) explored numerical methods, including finite volume, finite difference, and fractional derivatives, confirming calculus as the backbone of computational fluid dynamics (CFD). Only two studies (11%) looked at advanced approaches, particularly the use of fractional-order calculus in flight dynamics and control. Their findings suggest that these methods can strengthen stability when aircraft operate under nonlinear or rapidly changing conditions. Taken as a whole, the review indicates that calculus is not just a theoretical tool but also a means of driving progress in aerodynamics. Even so, gaps remain: few works explore real-time applications in autonomous flight, integration with machine learning is still limited, and fractional calculus has seen little use in practice. Filling these gaps depends on integrating mathematical models, computational techniques, and engineering practice in order to advance safer and more efficient aerospace applications.