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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	1. Acquire knowledge of and proficiency in the techniques of differential and integral calculus of real functions of several real variables — essential tools in the exact and applied sciences. 2. Improve graphic perception and three-dimensional visualisation by refining geometric skills. 3. Master the concepts, methods of resolution and fundamental results of ordinary differential equations, as well as their applications in mathematical modelling.
Syllabus	1. Geometry and topology 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's formula. 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. 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
Teaching Methodologies and Assessment Criteria	1. Assessment can take place during the teaching-learning period (continuous assessment) or in a final exam. All students are admitted to exam. 2. Continuous assessment requires 70 per cent class attendance, except for working students and finalists. 3. Attendance will be recorded from 26 February 2024. 4. The assessment of knowledge throughout the teaching-learning period will be periodic and will consist of two written tests, each lasting 2 hours and worth 10 marks, to be held on 22 April 2024 and 7 June 2024. 5. Students with a minimum of 3 marks in each written test and a sum of the marks in the two tests equal to or greater than 9.5 in the continuous assessment will be exempt from the final exam. 6. The use of calculators in assessment tests is not permitted. 7. Any attempt to cheat during an assessment test will fail the course unit Calculus II.
Language	Portuguese. Tutorial support is available in English.