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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 continuity1.1. Brief notions of topology in Rn1.2. Vectors. Norm of a vector. Scalar product. Velocity1.3. Vectorial functions and real multivariable functions1.4. Limits and continuity2. Differential Calculus in Rn2.1. Partial derivatives and directional derivatives2.2. Differentiability of functions from Rn into Rm2.3. Chain rule2.4. Derivatives of a higher order; Schwarz’s theorem2.5. Implicit function theorem2.6. Local and absolute extreme values2.7. Extremes with constraints: Lagrange multipliers3. Integral Calculus in Rn3.1. Riemann integral: definition and examples3.2. Properties of integrable functions3.3. Change of coordinates 3.4. Applications4. Ordinary Differential EquationsDefinition, 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.

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Bioengineering
Last updated on: 2024-03-18

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