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Syllabus |
1. Topology and Geometry in R^n.
Inner product, norm, and distance. Cross product. Lines and planes. Quadratic forms. Basic topological notions.
2. Differential Calculus in R^n.
Scalar- and vector-valued functions.
Graphs and level sets. Continuity. Partial and directional derivatives. Tangent plane. Differentiability. Higher-order derivatives. Derivative of a composite function (chain rule). Taylor’s formula. Local and constrained extrema. Implicit Function Theorem.
3. Multiple Integrals.
Definition, Fubini’s theorem, and change of variables.
Double integrals: Cartesian and polar coordinates.
Triple integrals: Cartesian, cylindrical, and spherical coordinates. Applications.
4. Vector Calculus.
Line and surface integrals.
Green’s theorem, Stokes’ theorem, and Gauss’ theorem.
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Teaching Methodologies and Assessment Criteria |
Classes are delivered in a combined lecture–practical format. The teacher presents the concepts and results and illustrates the theory through examples and applications. Students are encouraged to take an active part in class, interacting with the instructor and classmates, contributing to discussions of the topics, formulating and solving problems, and working through exercises. Independent study is also encouraged.
ASSESSMENT CRITERIA
1. Assessment may take place during the teaching period or by final examination.
2. Continuous assessment throughout the teaching–learning period will be carried out periodically and will consist of two written tests, each lasting two hours and worth ten (10) points, to be held on 21 April 2026 and 5 June 2026.
3. Students who obtain a mark of 9.5 points or higher in the continuous assessment carried out during the teaching period will be exempt from the final exam.
4. The final exam consists of a written paper worth twenty (20) points.
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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] Marsden and Tromba, Vector Calculus, 6th Edition, W.H. Freeman, 2011
[3] Salas, Hille, Etgen, Calculus: One and Several Variables, 6th Edition, John Wiley & Sons, Inc, 2007
[4] Adams, R., Essex, C., Calculus, A Complete Course, 9th Edition, Pearson, 2018
[5] Kreyszig, Advanced Engineering Mathematics, 10th Edition, John Wiley & Sons, Inc, 2011
[6] Anton, H., Bivens, I., Cálculo, Volume 2, Stephen Davis, 8.ª Edição, Bookman, 2007
[7] Apostol, T., Cálculo, Volume 2, Reverté, 1994
[8] Pires, G., Cálculo Diferencial e Integral em R^n, IST Press, 2012
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