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.

Series of exercises will be proposed with the aim of consolidating knowledge and stimulating problem-solving skills. The evaluation of the discipline, expressed on a scale from 0 to 20 points, will be made at different times, including 2 midterms (40% + 50%) and individual work to be developed outside the classroom (10%). If the weighted average of these evaluations is equal to or greater than 9.5, the student will be successful in the subject, otherwise the student will be able to attend a global frequency. In the final exam, the student can improve the grade. The minimum passing grade for these assessments is also 9.5. Assessment criteria are explained at the beginning of the semester.

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.