Calculus II

Code	10346
Year	1
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Does not have.
Mode of delivery	Theoretical/Practical
Work placements	Not applicable.
Learning outcomes	In this Curricular Unit the students will obtain the basic knowledge of series and of differential and integral Calculus of functions of several variables. At the end of this Curricular Unit the students should - determine the nature of a numerical series; - compute the interval of convergence of a power series; - compute limits of functions of several variables; - study the continuity of functions of several variables; - compute partial derivatives of functions of several variables; - apply the derivatives to compute maximums and minimums; - integrate functions of several variables; - apply integrals to compute areas and volumes; - compute line integrals; - apply Green's Theorem; - compute surface integrals; - apply Gauss and Stokes's theorems.
Syllabus	1. NUMERICAL AND POWER SERIES Nature of a series Comparison, limit, D'Alembert, Cauchy and Leibniz's tests Absolute convergence Power series 2. FUNCTIONS FROM R^n TO R^m Topological notions Functions Limits Continuity 3. DIFFERENTIAL CALCULUS IN R^n Partial derivatives Gradient, laplacian, jacobian, divergence and rotational Derivative in a point in the direction of a vector Differentiabiity Tangent hyperplane Linear approximation Chain rule Schwarz and implicit function's theorems Local and global extremes Lagrange multipliers 4. INTEGRAL CALCULUS in R^n Riemann integral Fubini's theorem Integrals in general sets Polar, cylindrical and spherical coordinates Areas and volumes 5. LINE INTEGRALS Line integrals of scalar and vectorial fields Green's theorem 6. SURFACE INTEGRALS Surface integrals of scalar and vectorial fields Gauss and Stokes's theorems
Main Bibliography	Main bibliography - Apostol, T.M., Cálculo, Vol. 1 e 2, Reverté, 1993 - Sarrico, C., Cálculo Diferencial e Integral, Esfera do Caos, 2009 - Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 2, McGrawHill, 1983 Additional bibliography - Dias Agudo, F.R., Análise Real, Vol. I e II, Escolar Editora, 1989 - Demidovitch, B., Problemas e exercícios de Análise Matemática, McGrawHill, 1977 - Lima, E. L., Análise Real, Vol. 2, Colecção Matemática Universitária, IMPA, 2004 - Lima, E. L., Curso de Análise, Vol. 2, Projecto Euclides, IMPA, 1989 - Mann, W. R., Taylor, A. E., Advanced Calculus, John Wiley and Sons, 1983 - Stewart, J., Calculus (International Metric Edition), Brooks/Cole Publishing Company, 2008
Teaching Methodologies and Assessment Criteria	The assessment will consist in two tests, each of one with a maximum value of 10 points. Designating the result of the first test by T1 and the result of the second test by T2, the final grade will be calculated as follows: - if T1 + T2 is less than 18.5 points, the final grade will be the rounding of T1+T2; - if T1 + T2 is greater than or equal to 18.5 points, the student must do an oral examination; the result of the oral exam, that we will designate by PO, will be between 0 and 20 points; the final grade will be the rounding of max {18, (T1 + T2 + PO) / 2}. The students with a final grade greater than or equal to 10 points will pass the course. All students will be admited to exam.
Language	Portuguese. Tutorial support is available in English.