| 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 |
Calculus I.
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Learning outcomes |
Problem solving and interpretation of results involving ordinary differential equations.
Domain of differential and integral calculus of scalar functions of several real variables.
At the end of this course, students should be able to:
1. Solve first and second order ordinary differential equations;
2. Formulate and solve problems modeled with ordinary differential equations;
3. Compute limits, study the continuity and differentiability of real functions with several real variables;
4. Find and classify free and constrained extrema;
5. Compute multiple integrals and apply the variable change theorem;
6. Compute areas and volumes using multiple integrals.
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Syllabus |
1. Ordinary Differential Equations.
1.1. Definition. Order and degree. Solution. Initial value problem.
1.2. First-order differential equations: separable equations, linear differential equations.
1.3. Second-order linear differential equations with constant coefficients.
1.4. Applications.
2. Differential Calculus in R^n
2.1. Algebra and Topology of R^n.
2.2. Real functions of several real variables.
2.3. Limits and Continuity.
2.4. Partial derivatives. Directional derivatives. Gradient.
2.5. Tangent plane. Linear approximation.
2.6. Differentiability.
2.7. The chain rule.
2.8. Higher-order derivatives. Schwarz Theorem.
2.9. Implicit function differentiation.
2.10. Maximum and minimum values.
2.11. Constrained extrema and Lagrange multipliers.
3. Integral Calculus in R^n
3.1. Multiple integrals: definition, examples, and properties. Fubini’s Theorem.
3.2. Change of variables.
3.3. Applications. Areas and volumes.
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Main Bibliography |
[1] Cálculo, Volume 2, James Stewart, Tradução da 7.ª edição norte-americana, 2014, Cengage Learning Edições Ltda
[2] Cálculo, Volume 2, Howard Anton, Irl Bivens, Stephen Davis, 8.ª Edição, 2007, Bookman
[3] Cálculo Diferencial e Integral para Funções de Várias Variáveis, Carlos Sarrico, 2009, Esfera do Caos
[4] Cálculo, Volume 2, T. Apostol,1994, Reverté
[5] Vector Calculus, J. Marsden, A. Tromba, 2003, Freeman and Company
[6] Cálculo Diferencial e Integral em R^n, Gabriel E. Pires, 2012, IST Press
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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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