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Calculus II

Calculus II

Code	16667
Year	1
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	-
Learning outcomes	The main goal of the Calculus II course is to develop skills and knowledge in differential and integral calculus of real functions of several real variables, providing the student with a range of essential tools in areas of exact or applied sciences. It is also intended to develop graphical perception and three-dimensional visualization by improving geometric skills and introducing fundamental concepts and techniques on ordinary differential equations and their applications in mathematical modelling.
Syllabus	1. Topology and geometry in Rn. Inner product, norm, and distance. Exterior product. 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 of IVP. Separation of variables and linear equations. Modelling with ODEs. Using power series and Laplace transforms for solving ODEs. Euler’s method.
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 [9] Boyce, W., DiPrima, R., Elementary Differential Equations and Boundary Value Problems, 10th Edition, John Wiley & Sons, Inc, 2012
Language	Portuguese. Tutorial support is available in English.