| Code |
14355
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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 |
-
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Learning outcomes |
The main objectives of the curricular unit of Calculus II are: to develop skills and comprehension in differential and integral calculus in several real variables, providing the student with a range of essential tools on exact or applied sciences; to develop graphical perception and three-dimensional visualization; to introduce fundamental concepts and techniques on ordinary differential equations and their applications in mathematical modelling.
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Syllabus |
Ordinary differential equations.
First order differential equations: slope fields and integral curves; existence and unicity. Separation of variables and linear equations. Modelling with ODEs. Using power series and Laplace transforms for solving ODE. Euler’s method.
Topology and geometry in Rn.
Inner product, norm and distance. Exterior product. Polar, cylindrical and spherical coordinates. Lines and planes. Quadratic forms. Topological notions.
Differential calculus in Rn.
Scalar and vector functions. Graphs and level sets. Continuity. Partial derivatives and directional derivatives. Tangent plane. Differentiability. Higher order derivatives. Gradient. The chain rule. Taylor series. Local and constrained extrema. Implicit function theorem.
Multiple integrals.
Definition, Fubini’s theorem and change of variables. Double integrals: rectangular and polar coordinates. Triple integrals: rectangular, cylindrical and spherical coordinates. Applications.
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Main Bibliography |
[1] Stewart, J., Cálculo, Volume 2, Tradução da 7.ª edição norte-americana, Cengage Learning Edições Ltda, 2014
[2] Apostol, T., Cálculo, Volume 2, Reverté, 1994
[3] Boyce, W., DiPrima, R., Elementary Differential Equations and Boundary Value Problems, 10th Edition, John Wiley & Sons, Inc, 2012
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Teaching Methodologies and Assessment Criteria |
1. The teaching methodology is based on theoretical and practical lessons. The theoretical part is based on the teacher's presentation of the syllabus contents, based on the bibliography of the unit or other notes available. Great focus will be given to the rigorous demonstration of the main results. The practical part of the classes is based on solving exercises, both in an accompanying and autonomous way.
2. The assessment is done through two written tests, carried out in the middle and at the end of the semester.
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Language |
Portuguese. Tutorial support is available in English.
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