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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 During the teaching-learning period, the evaluation will be periodic and will consist of 2 tests to be carried out on April 28 and June 9, each with a quotation of 10 values. A student will pass if the sum of the test scores has a rating greater than or equal to 9.5. Whenever the final classification is greater than or equal to 17 values, a defense will be held, in any case a minimum grade of 16 values ??will be assured. If, in the event of a defense, the student does not appear, he/she will have a final grade equal to 16 values.

All students are admitted to the exam.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2022-06-15

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