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Calculus II

Code 16667
Year 1
Semester S2
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements -
Learning outcomes 1. Acquire knowledge of and proficiency in the techniques of differential and integral calculus of real functions of several real variables — essential tools in the exact and applied sciences.
2. Improve graphic perception and three-dimensional visualisation by refining geometric skills.
3. Master the concepts, methods of resolution and fundamental results of ordinary differential equations, as well as their applications in mathematical modelling.


Syllabus 1. Geometry and topology in Rn.
Inner product, norm, and distance. Exterior product. Lines and planes. Quadratic forms. Topological notions.
2. Differential calculus in Rn.
Scalar and vector functions. Graphs and level sets. Continuity. Partial derivatives and directional derivatives. Tangent plane.
Differentiability. Higher order derivatives. Gradient. The chain rule. Taylor's formula. Local and constrained extrema. Implicit function theorem.
3. Multiple integrals.
Definition, Fubini’s theorem, and change of variables. Double integrals: rectangular and polar coordinates. Triple integrals: rectangular, cylindrical, and spherical coordinates. Applications.
4. Ordinary differential equations.
First order differential equations: slope fields and integral curves; existence and unicity of IVP. Separation of variables and linear equations.
Modelling with ODEs. Euler’s method.
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] Salas, Hille, Etgen, Calculus: One and Several Variables, 6th Edition, John Wiley & Sons, Inc, 2007
[4] Adams, R., Essex, C., Calculus, A Complete Course, 9th Edition, Pearson, 2018
[5] Kreyszig, Advanced Engineering Mathematics, 10th Edition, John Wiley & Sons, Inc, 2011
[6] Anton, H., Bivens, I., Cálculo, Volume 2, Stephen Davis, 8.ª Edição, Bookman, 2007
[7] Apostol, T., Cálculo, Volume 2, Reverté, 1994
[8] Pires, G., Cálculo Diferencial e Integral em R^n, IST Press, 2012
[9] Boyce, W., DiPrima, R., Elementary Differential Equations and Boundary Value Problems, 10th Edition, John Wiley & Sons, Inc, 2012
Teaching Methodologies and Assessment Criteria Classes are theoretical and practical. The teacher presents the concepts and results and illustrates the theory with examples and applications. Students are encouraged to participate in class, interacting with the teacher and classmates, contributing to discussions on the topics, formulating and solving problems, and completing exercises. Independent work is also encouraged.

ASSESSMENT CRITERIA
1. Assessment may be carried out during the course or in a final examination.
2. Knowledge assessment throughout the teaching-learning period will be periodic and will consist of two written tests, each lasting two hours and worth ten (10) points, to be held on 21 April 2026 and 5 June 2026.
3. Students who have obtained a grade of 9.5 or higher in the assessment carried out throughout the teaching activities will be exempt from the final exam.
4. The final exam consists of a written test graded out of twenty (20) points.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2026-02-27

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