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 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., & 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.