MATHEMATICAL MODELING OF DYNAMIC PHENOMENA: CONTRIBUTIONS OF DIFFERENTIAL CALCULUS TO THE TEACHING OF MATHEMATICS AND PHYSICS
DOI:
https://doi.org/10.66104/xf7nd968Keywords:
Differential Calculus. Derivative. Wave Mechanics. Classical Physics. Kinematics.Abstract
This article examines the foundations of differential calculus and its applications to central problems in Classical Physics, with emphasis on kinematics and wave mechanics. Beginning from the rigorous limit-based definition of the derivative, as formalised by Cauchy and Weierstrass in the nineteenth century, the work develops the operational rules of differentiation, discusses the geometric and physical interpretation of the derivative, and analytically demonstrates the equations of uniformly accelerated motion. Subsequently, the differential operator is applied to the sinusoidal progressive wave equation, yielding explicit expressions for the instantaneous velocity and acceleration of a point in the vibrating medium. The mathematical treatment shows that the phase velocity of a transverse mechanical wave on a tensioned string is governed jointly by the elastic property (tension T) and the inertial property (linear mass density µ), producing the expression v = √(T/µ), which is confirmed by dimensional analysis. The methodology is analytical-deductive, employing the formal apparatus of differential calculus and Leibniz notation. Results demonstrate that differentiation constitutes an indispensable tool for the precise modelling of dynamic physical phenomena, transcending the limitations of elementary algebraic approaches
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Copyright (c) 2026 Francisco Arlon de Oliveira Chaves Oliveira, Francisca das Chagas Oliveira, Evandro de Carvalho Ribeiro, Eugenia Maria dos Santos Cordeiro, Andreson de França Almeida, Gilvan Moreira da Paz
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