| Code |
15965
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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 |
Classes are predominantly theoretical and practical. They are expository, including concepts, fundamental results, demonstrations, examples, and applications to other Sciences. The students' participation is also encouraged and some practical classes are held to solve exercises, in work groups or individually, with the teacher's guidance.
The assessment can be carried out during the class period and consists of two small and two written tests, or in a final exam.
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Language |
Portuguese. Tutorial support is available in English.
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