| 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 |
Real Analysis and Linear Algebra.
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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 |
- Theoretical-practical classes and homework;
The assessment will be based on the following elements:
1. Three tests, T1, T2 and T3, all quoted for 20 points.
2. Two mandatory assignments/exercises at home, E1 and E2, both priced at 10 points.
The final classification F, rounded to the nearest unit, is given by the formula F=0.3*T1+0.35*T2+0.3*T3+0.05*(E1+E2)
The student will pass Continuous Assessment in the curricular unit if the classification in each assessment element is equal to or greater than 5 values ??and the final classification F is equal to or greater than 9.5 values. The approved student will have a classification equal to F.
All students will be admitted to the exam.
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Language |
Portuguese. Tutorial support is available in English.
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