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Calculus II

Code 9090
Year 1
Semester S2
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements Not applicable
Mode of delivery Face to Face with recourse to e-learning.
Work placements N/A
Learning outcomes It is intended that the students develop a clear understanding, relate and apply the fundamental concepts of multivariable calculus:
1. Study vector functions and functions of several variables: domains, ranges, graphs, level sets, topology in Rn;
2. Compute limits and study continuity;
3. Compute partial derivatives and study differentiability;
4. Recognize the significance of the gradient and its relationship with directional derivatives and linear approximation;
5. Apply chain rule and the implicit function theorem;
6. Set up and solve optimization problems with or without constraints;
7. Set up and compute multiple integrals in cartesian, polar, cylindrical, and spherical coordinates, using double and triple integration, including use of change of variables techniques;
8. Apply multiple integrals to compute areas, volumes, mass, density function, work, and fluxes.
9 Solve basic differential equations.
10 Apply basic differential equations to mathematical modelling.
Syllabus 1. Functions from Rn into Rm: limits and continuity
1.1. Brief notions of topology in Rn
1.2. Vectors. Norm of a vector. Scalar product. Velocity
1.3. Vectorial functions and real multivariable functions
1.4. Limits and continuity
2. Differential Calculus in Rn
2.1. Partial derivatives and directional derivatives
2.2. Differentiability of functions from Rn into Rm
2.3. Chain rule
2.4. Derivatives of a higher order; Schwarz’s theorem
2.5. Implicit function theorem
2.6. Local and absolute extreme values
2.7. Extremes with constraints: Lagrange multipliers
3. Integral Calculus in Rn
3.1. Riemann integral: definition and examples
3.2. Properties of integrable functions
3.3. Change of coordinates
3.4. Applications
4. Ordinary Differential Equations
Definition, examples and applications. Method of separation of variables. Homogeneous equations. Exact equations. Linear differential equations, Integrating factor. Equations of Bernoulli, Ricatti and Clairaut.
Main Bibliography [1] Cálculo, vol. II, James Stewart, 2006, Pioneira Thomson Learning
[2] Cálculo, vol. 2, Howard Anton, Irl Bivens, Stephen Davis, 8ª Edição, 2007, Bookman
[3] Análise Real, vol.2 - Funçoes de n Variaveis, Elon Lages Lima, Coleçao Matematica Universitaria, IMPA (Brasil), 2007.
[4] Análise Real, vol.3 - Analise Vetorial, Elon Lages Lima, Coleção Matemática Universitária, IMPA (Brasil), 2007.
[5] Vector Calculus, J. Marsden, A. Tromba, 2003, Freeman and Company.
[6] Cálculo, vol. II, T. Apostol,1994, Reverté.
Teaching Methodologies and Assessment Criteria The curricular unit runs in theoretical-practical type of lessons. In the first part of the lesson, the professor exposes the most relevant results and techniques, usually illustrated by examples. The second part of the lesson is devoted to problem solving by the students. The students are proposed to solve a list of problems from the adopted book.
Three written tests will take place: T1, T2, and TG.
The student is approved if the classification “Ensino-Aprendizagem” or the classification in one of the exams is greater or equal than 10 points.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2024-03-18

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