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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 The main goal of the Calculus II course is to develop skills and knowledge in differential and integral calculus of real functions of several real variables, providing the student with a range of essential tools in areas of exact or applied sciences. It is also intended to develop graphical perception and three-dimensional visualization by improving geometric skills and introducing fundamental concepts and
techniques on ordinary differential equations and their applications in mathematical modelling.
Syllabus 1. Topology and geometry 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 series. 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. Using power series and Laplace transforms for solving ODE. 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 The classes have a theoretical and practical nature. They have an expository character: fundamental results, examples, and applications. Student participation is emphasized, and practical classes are conducted for solving exercises individually or in groups.
Students can pass Calculus II either through the teaching-learning process or by examination. The minimum grade required to pass Calculus II is 9.5.
Students will be eligible for the exam if they have a Teaching-Learning Classification (CEA) of at least 5, as well as working and final-year students.
Students will be exempt from the final exam if they obtain a CEA of 9.5 or higher.
Students with a CEA of 9.5 or higher, but less than or equal to 17, are considered approved. They may take a supplementary test If they obtain a CEA or an exam grade higher than 17.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2025-03-21

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