Class Calculus II

Calculus II develops the fundamental concepts of multivariable mathematical analysis and constitutes a core component of Engineering education. The course covers the study of curves and surfaces, scalar and vector-valued functions of several variables, differentiation and optimization, as well as multiple integration and line integrals, providing the mathematical foundations for modelling and analysing multidimensional problems. The course equips students with essential tools for the modelling and analysis of multidimensional phenomena, fostering logical reasoning, abstract thinking, and mathematical problem-solving skills, while providing the necessary background for subsequent courses.

Not applicable

Basic notions in R^n. Vector, geometrical and topological structure. Curves and paths. Continuity and differentiability. Regular paths. Reparametrizations. Functions of several real variables. Conics and quadrics. Domain. Level sets. Limits. Sandwich theorem. Properties of limits. Continuity. Partial derivatives. Derivatives. Differentiability. Gradient. Hessian Matrix. Taylor's Polynomial. Extremes. Critical Points. Multivariable functions. Domain. Level sets. Limits and continuity. Differentiability. Derivatives. Jacobian Matrix. Chain rule. Line integrals of scalar fields. Invariance by reparametrization. Length of curves. Line integral of vector fields. Signal and reparametrization invariance. Work. Fundamental theorem of the calculus. Conservative vector fields. Double integrals. Fubini's Theorem. Change of variables. Green's Theorem. Applications.

Series of exercises will be proposed to consolidate knowledge and stimulate problem-solving skills. 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., & Davis, S. (2012). Calculus (10th ed.). John Wiley & Sons. Sarrico, C. (2009). Cálculo Diferencial e Integral para Funções de Várias Variáveis. Esfera do Caos. Stewart, J., Clegg, D., & Watson, S. (2020). Calculus (9th ed.). Cengage.