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.
Language	Portuguese. Tutorial support is available in English.