Calculus II

Code	15851
Year	1
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Not applicable
Mode of delivery	Lectures and practical classes with face-to-face teaching.
Work placements	Not applicable.
Learning outcomes	This course constitutes an introduction to differential and integral calculus in R^n. At the end of this curricular unit, the student must be able to: a) Solve some first and second-order ordinary differential equations b) Compute limits of functions of several variables. c) Investigate the continuity of functions of several variables. d) Compute the derivatives of functions of several variables. e) Apply the derivatives to compute maximums and minimums. f) Integrate functions of several variables. g) Apply integral calculus to compute areas and volumes. h) Solve line integrals.
Syllabus	1. Ordinary differential equations: separable equations; first-order linear differential equations; second-order linear differential equations with constant coefficients 2. Functions from R^n into R ^m: Brief notions of topology in R^n; Functions from R^n into R^m; Limits; Continuity 3. Differential Calculus in R^n: Partial derivatives and directional derivatives; Differentiability of functions from R^n into R^m; Derivatives of higher order; Schwarz’s theorem; Local and absolute extreme values; Extremes with constraints: Lagrange multipliers 4. Integral Calculus in R^n: Riemann integral: definition and examples; Properties of integrable functions; Change of coordinates; Applications 5. Line integrals: parameterization of curves; line integrals of scalar functions (length); line integrals of vector fields; Green's Theorem
Main Bibliography	- Apostol, T.M., "Cálculo", Volume 1 e 2, Reverté, 1993. - Azenha, A., Jerónimo, M.A., Elementos de Cálculo Diferencial e Integral em R e Rn, McGraw-Hill, 1995 - Stewart, J., Cálculo, Volumes I e II, Tradução da 6a edição Norte-Americana, Cengage Learning, 2010. - Course Slides (Moodle)
Language	Portuguese. Tutorial support is available in English.