This Course Unit aims to give an introduction to the study of mathematical analysis in the R^n space and to endow the student with the basic tools for calculating on this space.
At the end of the course the student should be able to:
1. To analyze in detail a function of several variables, regarding continuity, differentiability and relative extrema.
2. To define scalar fields, vector fields and conservative fields.
3. To compute and interpret certain differential operators in scalar and vector fields.
4. To understand the geometric idea of directional derivative, the double and triple integral and also of line integrals and surface integrals.
5. To calculate areas and volumes using double and triple integrals, as well as curvilinear and surface integrals of scalar and vector fields.
1. Functions Series
1.1. Power Series
1.2. Taylor's Serie
2. Basic notions of geometry and topology in R^n: inner product, norm and distance. Open and closed sets of R^n. Limited Sets.
3. Real Functions of several real variabes
3.1. Vector Functions: domain and graphic, level curves and surfaces.
4. Differential Calculus in R^n
4.1. Diferentiability of functions with several variables: partial derivatives, tangent plan, normal line, Schwarz's theorem and chain rule.
4.2 Applications: local extremes and constaint extremes.
5. Integral Calculus in R^n
5.1. Multiple Integrals: definition and properties.
5.2. Double Integral: geometric interpretation, change of variables (polar coordinates) and applications.
5.3. Triple Integrals: geometric interpretation, change of variables (cylindrical and spherical coordinates) and applications.
6- Surface Integrals
Alberto Simões, Cálculo II, Universidade da Beira Interior.
1. Stewart, James, "Cálculo", Volume II, 5ª edição Thomson Learning, 2001.
2. Lang, S., "Calculus of Several Variables", Undergraduate Texts in Mathematics, Third Edition, Springer-Verlag,1987.
3. Apostol,T.M., "Calculus",Volume II, John Wiley & Sons, 1968.
4. J. Marsden e A. Tromba, Vector Calculus, W H Freeman & Co., 2003.
5. Jaime Carvalho e Silva, Princípios de Análise Matemática Aplicada, Mc Graw Hill, 1999.
6. Cálculo diferencial e integral, vol. I e vol. II, N. Piskounov, Lopes da Silva, 1987.
7. Robert A. Adams, Calculus: A Complete Course, Addison-Wesley, 2006.
8. H. Anton, I. Bivens e S. Davis, Calculus, (Eight Edition), John Wiley & Sons, 2006.
Teaching Methodologies and Assessment Criteria
The classes will be theoretical-practical, that is, the contents will be presented to the students with the help of practical problems.
Concerning the continuous educational process: the final evaluation of this process will be obtained by the sum of the evaluations obtained in the two written tests, rounded to the units. Those students with ta classification equal or higher than 10 points will be dispensed of the final exam.