| Code |
14767
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| Year |
2
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| Semester |
S1
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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 |
N.A.
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Learning outcomes |
i) To understand, to relate and to apply concepts and basic results of integral calculus;
ii) To formulate and to solve problems related to multiple integrals, line integral and surface integrals;
iii) To solve problems related to integrals of differential forms;
iii) To analyze and understand mathematical proofs;
iv) To communicate using mathematical language, written and orally.
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Syllabus |
1. Multiple integrals
1.1. Construction of the integral and properties
1.2. Null measure sets and integrability
1.3. Fubini’s theorem
1.4. Change of coordinates
1.5. Application to the calculus of physical measures
2. Line integrals
2.1. Parametrization of curves
2.2. Line integrals of scalar and vector fields
2.3. Green’s theorem
3. Surface integrals
3.1. Differentiable surfaces
3.2. Integrals of scalar fields and flows of vector fields
3.3. Stokes’ curl theorem
3.4. Gauss-Ostrogradsky’s divergence theorem
4. Integrals of differential forms
4.1. Differential forms of degree 1
4.2. Line integral of a differential form
4.3. Invariance by homotopy
4.4. Closed and exact forms. Poincaré’s Lemma
4.5. Exterior product and differential forms of degree 2. Exterior differential
4.6. Surface integral of a differential form
4.7. Stokes’ theorem
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Main Bibliography |
Vector Calculus, P. Baxandall & H. Liebeck. Dover, 2008
Calculus of Several Variables, S. Lang, Second Edition, Addison-Wesley Publishing Company.
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Teaching Methodologies and Assessment Criteria |
The classes will be theoretical-practical. The teacher presents the concepts and enunciates the results, demonstrating many of them. It also illustrates the theory with examples and applications. The student is encouraged to participate in classes, interacting with the teacher and sometimes solving exercises and problems. Autonomous work will be encouraged, mainly consisting of exercises, problem-solving and mathematical demonstrations. The evaluation carried out over the teaching-learning period will consist of three written tests MT1 (10/10/2022), MT2 (7/12/2022) and FT (12/12/2022) (0-20 values). The final classification will be given by FC =Maximum(0,25*MT1+0,25*MT2+0,5*FT,FT). The student can also take a final exam.
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Language |
Portuguese. Tutorial support is available in English.
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