Calculus II

Code	12809
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	1. It is intended that the students develop a clear understanding of the main tools in the multivariable calculus. The student should be able to analyze vector functions and scalar functions of several variables. Specifically, the students shall: Compute limits and study continuity; Compute partial derivatives and study differentiability; Recognize the significance of the gradient and its relationship with directional derivatives and linear approximation; Apply the chain rule and the implicit function theorem; Set up and solve optimization problems with or without constraints; Compute multiple integrals, including use of change of variables techniques, identify the coordinate systems and make change of variable; 2. It is intended that the students develop a clear understanding of the basic results on ordinary differential equations. The student should be able to solve basic differential equations and to apply basic differential equations to mathematical modelling.
Syllabus	1. Functions from Rn into Rm: 1.1. Vectorial functions and real multivariable functions 1.2. 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. Numerical methods: Euler method.
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.