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

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.

A series of exercises will be proposed to consolidate students' knowledge and develop their problem-solving skills. In each class, students are encouraged to work on questions related to the topics covered and to present the results of their individual work in the following session. Throughout the course, particular emphasis is placed on the fundamental idea that solving exercises is not an end in itself, but rather a means of achieving a deeper understanding of the underlying mathematical concepts. Mastery of these concepts enables students to apply the acquired techniques to the solution of more advanced engineering and scientific problems. To further support learning and promote student engagement, short videos on selected course topics, historical notes, mathematical curiosities, recreational problems, and real-world applications will be made available throughout the course.

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.