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Calculus II

Code 16140
Year 1
Semester S2
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements N.A.
Learning outcomes The students approved in this course will be able to:
1. Compute limits of functions of several variables
2. Investigate the continuity of functions of several variables
3. Investigate the differentiability of functions with several variables
4. Apply the derivatives to compute maximuns and minimuns
5. Integrate functions of several variables
6. Apply integral calculus to compute areas and volumes
7. Compute line and surface integrals
Syllabus 1. Improper integrals
2- Real functions of several real variables: limits and continuity
2.1 Basics in R^n: algebra, geometry and topology
2.2 Domain, range, graph, curves and level surfaces
2.3 Limits
2.4 Continuity
3- Differential Calculus in R^n
3.1 Partial derivatives and directional derivatives
3.2 Differentiability
3.3 Derivative of the composite function
3.4 Higher order derivatives; Schwarz's Theorem
3.5 Implicit Function Theorem
3.6 Local extremes and absolute extremes
3.7 Conditioned extremes: Lagrange multipliers
4- Integral calculus in R^n
4.1 Double and triple integrals: definition, examples and properties
4.2 Change of coordinates
4.3 Applications
5- Line integrals
5.1 Paths and lines
5.2 Line integral of a scalar field
5.3 Line integral of a vector field4.4 Green's Theorem
Main Bibliography [1] Stewart, J., Cálculo, Volume 2, Tradução da 7.ª edição Norte-Americana, Cengage Learning Edições Ltda, 2014
[2] Marsden and Tromba, Vector Calculus, 6th Edition, W.H. Freeman, 2011
[3] Adams, R., Essex, C., Calculus, A Complete Course, 9th Edition, Pearson, 2018
[4] Anton, H., Bivens, I., Cálculo, Volume 2, Stephen Davis, 8.ª Edição, Bookman, 2007
[5] Apostol, T., Cálculo, Volume 2, Reverté, 1994
[6] Pires, G., Cálculo Diferencial e Integral em R^n, IST Press, 2012
[7] Kreyszig, Advanced Engineering Mathematics, 10th Edition, John Wiley & Sons, Inc, 2011
[8] Salas, Hille, Etgen, Calculus: One and Several Variables, 6th Edition, John Wiley & Sons, Inc, 2007.
Teaching Methodologies and Assessment Criteria 1. Classes are theoretical-practical. The teaching method consists of the exposition of concepts, theoretical results, examples and applications to engineering, whenever possible. Problem solving and individual work are encouraged, based on exercises suggested by the teacher.

2.1. The assessment can take place during the class period or in the final exam.
2..2. The assessment of knowledge throughout the teaching-learning period will be periodic and will consist of two written tests, each lasting two hours and rated at ten (10) points.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2024-03-01

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