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Computational Mathematics II

Computational Mathematics II

Code	13204
Year	1
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Computational Mathematics I
Mode of delivery	Face to face.
Work placements	Not applicable.
Learning outcomes	The main objectives 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 modeling.
Syllabus	1. Topology and geometry in Rn. Inner product, norm and distance. Exterior product. Polar, cylindrical and spherical coordinates. Lines and planes. Quadratic forms. Topological notions. 2. 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. 3. Multiple integrals. Definition, Fubini’s theorem and change of variables. Double integrals: rectangular and polar coordinates. Triple integrals: rectangular, cylindrical and spherical coordinates. Applications. 4. Ordinary differential equations First order differential equations: slope fields and integral curves; existence and unicity. Separation of variables and linear equations. Modeling with ODE. Using power series and Laplace transforms for solving ODE. Euler’s method.
Main Bibliography	[1] Marsden and Tromba, Vector Calculus, 6th Edition, W.H. Freeman, 2011 [2] Salas, Hille, Etgen, Calculus: One and Several Variables, 6th Edition, John Wiley & Sons, Inc, 2007 [3] Stewart, J., Cálculo, Volume 2, Tradução da 7.ª edição norte-americana, Cengage Learning Edições Ltda, 2014 [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
Language	Portuguese. Tutorial support is available in English.