Class Calculus III

Calculus III deepens fundamental concepts of Mathematical Analysis and Vector Calculus essential to Engineering education. It covers sequences and series, power and Taylor series, ordinary differential equations, and vector calculus, including line and surface integrals and the main integral theorems. The course provides mathematical methods for modelling and analysing physical and technological phenomena, enabling the study of processes such as growth, heat transfer, vibrations, fluid flow, and force fields. It provides a foundation for advanced courses, including Mechanics, Thermodynamics, Fluid Mechanics, Electromagnetism, Control, and Numerical Methods, while developing analytical reasoning and the ability to solve engineering problems. Within the study programme, Calculus III consolidates the mathematical competences required for the modelling, analysis, and optimisation of engineering systems, strengthening the scientific background of future engineers.

Not applicable

Sequences. Series. Convergence. Geometric, Mengoli and Dirichelet series. Series of non-negative terms. Alternating series. Simple and absolute convergence. Leibniz criterion. Power Series. Convergence domain. Power series development. Taylor series. Ordinary Differential Equations (ODE). Simple, separable, linear ODE. Bernoulli's ODE. Growth models. Logistics models. Mixing and heating problems. Homogeneous differential equations. Second Order Ordinary Differential Equations. Vibrating models. Scalar and vector fields. Line integrals. Work done by a force. Independence of the path. Conservative fields. Green's theorem. Surface integrals. Divergence theorem. Stokes theorem.

In each session, students are encouraged to focus on questions related to the topics covered in class and to show the results of their individual work in the following session where questions are resolved on the board, whenever necessary, and details that have caused doubts are clarified. or difficulties. The fundamental idea is repeatedly highlighted that solving exercises has the main objective of allowing a deeper understanding of the conceptual body of the curricular unit, mastering which will allow the application of these tools to solve more advanced problems. Generally, a progression is adopted that starts from more calculative issues and develops towards the more conceptual parts of Calculus in Rn, thus gradually consolidating mastery of the various tools provided by this curricular unit.

Anton, H., Bivens, I. C., & Davis, S. Calculus . 11th ed. Hoboken, NJ: John Wiley & Sons, 2019. Sarrico, C. Cálculo Diferencial e Integral para Funções de Várias Variáveis . Lisboa: Esfera do Caos, 2009. Zill, D. G., & Wright, W. S. Differential Equations with Boundary-Value Problems . 10th ed. Boston, MA: Cengage Learning, 2017.