| Code |
12809
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| Year |
1
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| Semester |
S2
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Not applicable, but it is advisable to have prior knowledge of Calculus I (limits/continuity on R, differentiation and integration in 1D), basic notions of linear algebra/vectors, and fluent algebraic manipulation.
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Mode of delivery |
Face to Face with recourse to e-learning.
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Work placements |
N/A
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Learning outcomes |
The aim is for students to understand and apply differential and integral calculus in higher dimensions, as well as basic ODEs. In particular: (i) analyse functions Rn?R? with respect to limits and continuity; (ii) compute partial/directional derivatives, the gradient and tangent planes, verifying differentiability; (iii) apply the chain rule, higher-order derivatives, Schwarz’s theorem and the implicit function theorem; (iv) identify extrema (free and constrained, via Lagrange multipliers); (v) set up and solve double/triple integrals, perform changes of coordinates, and apply them to areas/volumes; (vi) model and solve separable/linear ODEs and second-order linear ODEs with constant coefficients. These objectives are articulated with TP classes: some focused on foundations and examples; others on guided problem-solving, Moodle tasks, assignments, and in-class presentations.
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Syllabus |
1. Ordinary Differential Equations
1.1. Definition, examples and applications. Separation of variables.
1.2. Linear equations, integrating factor method, and Bernoulli equations.
1.3. Second-order equations with constant coefficients.
1.4. Applications.
2. Functions from Rn to R?
2.1. Real-valued functions of n real variables and vector-valued functions.
2.2. Limits and continuity.
3. Differential Calculus in Rn
3.1. Partial derivatives. Directional derivatives. Gradient.
3.2. Tangent plane.
3.3. Differentiability.
3.4. Derivative of the composite function.
3.5. Higher-order derivatives. Schwarz’s theorem.
3.6. Implicit function theorem.
3.7. Local and absolute extrema.
3.8. Constrained extrema: method of Lagrange multipliers.
4. Integral Calculus in Rn
4.1. Double and triple Riemann integrals: definition and examples.
4.2. Properties of integrable functions.
4.3. Change of coordinates.
4.4. Applications.
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Main Bibliography |
[1]Calculus- A Complete Course, Robert Adams, Christopher Essex, 10ª edição, 2022, Pearson
[2] Cálculo, vol. II, James Stewart, 2006, Pioneira Thomson Learning
[3] Cálculo, vol. 2, Howard Anton, Irl Bivens, Stephen Davis, 8ª edição, 2007, Bookman
[4] Análise Real, vol.2 - Funções de n Variáveis, Elon Lages Lima, Coleção Matemática Universitária, IMPA (Brasil), 2007.
[5] Análise Real, vol.3 - Análise Vetorial, Elon Lages Lima, Coleção Matemática Universitária, IMPA (Brasil), 2007.
[6] Vector Calculus, J. Marsden, A. Tromba, 2003, Freeman and Company.
[7] Cálculo, vol. II, T. Apostol,1994, Reverté
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Teaching Methodologies and Assessment Criteria |
Continuous assessment (0–20):
CEA = max(0.1·AM + 0.2·MT + 0.7·Freqs, 0.1·AM + 0.9·Freqs),
where AM = Moodle activities (10%),
MT = average of 2 minitests (20%),
Freqs = average of 2 written tests (70%–90%).
Each written test includes the material assessed in the minitest immediately preceding it. Students may choose to answer the entire test or only the sections that exclude the minitest material.
The “non-penalisation” rule ensures that the presence of MT never reduces the final mark.
Examination assessment is independent of this model (0–20). For final marks above 17, a supplementary assessment is required to defend the grade.
Exempt from attendance and from the minimum grade required for exam access: working students, finalists, and students with special status.
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Language |
Portuguese. Tutorial support is available in English.
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