Mathematical Analysis II D
Code
10572
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
João Nuno Gonçalves Faria Martins
Weekly hours
6
Total hours
78
Teaching language
Português
Objectives
Understanding of the concept of a series of real numbers. Analysis of the convergence of a series of real numbers.
Understanding of the concept of limit and continuity in the context of real functions of several variables.
Study of the concept of differenciability for functions of several variables, with applications to the study of their extrema (absolute, relative and constrained). Lagrange multipliers.
Understanding of the concept of double and triple integrals, and their explicit calculation via Fubini theorem and changes of variables.
Understanding of line and surface integrals. Conservative fields. Green, Stokes and Gauss (divergence) theorems.
Prerequisites
Sequences of real numbers. Differential and integral calculus in one variable. Matrix calculus.
Subject matter
Numerical series
Definition and notion of convergence. The cases of geometric and telescoping series. Convergence criteria for series of non-negative terms. Absolute convergence. The Leibniz test for alternating series .
Limits and continuity for functions of several variables.
Overview of functions of several variables. Elementary topological notions in R^n . Notion of limit of a vectorial sequence. Limit of a real function of several variables. Relative and iterated limits. Using polar coordinates. The case of vector valued functions. Continuity.
Differential calculus for functions of several variables.
Differentiability of a function of several variables. Partial derivatives. Directional derivatives. Schwarz theorem. Derivative of composite functions. The case of vector valued functions. Implicit function theorem. Taylor formula. Relative and global extrema. Constrained extrema and Lagrange multipliers .
Multiple integration
Double and triple integrals. Fubini theorem. Changes of coordinates. Cylindrical and spherical coordinates Applications to the calculus of areas, volumes and centres of mass .
Line integrals
Line integrals of scalar and vector fields. Green''''s theorem. Gradient, curl, divergence. Differential forms. Conservative fields.
Surface integrals
Surface integrals. Flow of a vector field along a surface. Stokes theorem, Gauss theorem.
Bibliography
H. Anton, I. Bivens, S. Davis, Cálculo, volume 2, 8ª edição, Bookman, Porto Alegre, 2007.
G. E. Pires, Cálculo diferencial e integral em Rn, IST Press, Lisboa, 2012.
Carlos Sarrico. Cálculo Diferencial e Integral para funções de várias variáveis. Esfera do Caos Editores. 2009.
Calculus; Early Transcendentals-James Stewart (Sixth Edition). Capítulos 12 a 16.
Cálculo 2- Tom M Apostol, Editorial Reverté 1996. Capítulos 8 a 12.
Calculus; a New horizon-Howard Anton (Sixth Edition). Capítulos 12 a 17.
J. E. Marsden and A. Tromba, Vector Calculus, 5th ed., W. H. Freeman (2003).
Teaching method
Theoretical classes: fundamental concepts are taught, illustrated via numerous examples.
Practical classes: are devoted to solving exercises about the concepts introduced.
Extensive exercise sheets will be provided, of which the students should solve a large number before and after the classes (independently) and during the practical classes, with the help of a teacher.
Any doubts that arise may be answered during the classes and during the office hours, scheduled or arranged ad-hoc with the students, in case of time-table incompatibility.
In order to guide the students in the subject of the course, extensive notes of the theoretical classes will be provided. These do not substitute the presence in the classes.