The Role of Calculus in Mathematical Analysis of Physical and Engineering Systems review article

Abstract

The paper reviews differentiation and integration with regard to the analysis of physical and engineering systems in light of practical applications of state-of-the-art techniques. The research covers four major axes. First, it deals with basic theoretical concepts such as derivatives, integrals, and their relationship through the fundamental theorem of calculus. Second, it reviews the role of differential equations regarding the modeling of dynamic systems, such as motion and thermal diffusion. Third, the research addresses engineering and physical applications, such as designing dynamic structures and aerodynamics. Fourth, modern techniques involve the use of fractional calculus and differential-supported neural networks in order to enhance the accuracy and solve complex problems. Results indicate the importance of differentiation and integration as comprehensive tools of analysis applied for understanding natural and engineering systems; the role of modern techniques in enhancing their accuracy and efficiency is also pointed out. The study also demonstrates how differentiation and integration are adapted to be integrated with modern technologies, such as AI and nanotechnology, while increasing their importance in facing future challenges.