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Calculus II

Code 10274
Year 1
Semester S2
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements Do not exist
Mode of delivery Presential classes
Work placements Not applicable
Learning outcomes 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.
Syllabus 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.
3.2. Limits.
3.3. Continuity.
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
Main Bibliography 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 teaching-learning (TL) classification will be the best of A or B.
Option A: Test1 30%, Works 30%, Test2 40%;
Option B: Test1 50%, Test2 50%.
The tests will be performed using Zoom on the dates:
1st Test - May 6 from 6 pm to 8 pm; 2nd Test - June 5 from 6 pm to 8 pm.
The final classification (FC) will be: FC = E if TL <10; FC = max {TL, E} if TL >= 10
E being the note of the Normal Exam that will be done in person in a date and room to be defined by academic services.
All students enrolled in the Course are admitted to the Normal Exam and the Special Exam.
Registration for exams is mandatory in order to ensure safety conditions and adequate physical distance.
Registration for the Normal Exam must be sent by email to the teacher.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2020-06-30

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