Mathematical Analysis II C
Code
10347
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Nuno Filipe Marcelino Martins
Weekly hours
5
Total hours
69
Teaching language
Português
Objectives
At the end of this course students are expected to:
- have knowledge of the concepts, notations and objectives of Mathematical Analysis in R ^ n, especially for n = 2 and n = 3;
- are able to solve practical problems using derivatives and integrals of functions of several variables.
- have knowledge of the main theorems of differential and integral calculus, especially the theorems of Green, Stokes and divergence.
- know the notion of numerical series and know how to analyze the convergence of series of nonnegative real numbers and alternating series
Prerequisites
The students should have knowledge of linear algebra and analytic geometry, in particular of vector calculus in R ^ 2 and R ^ 3, the equations of lines and planes in R ^ 3 and basic matrix computations. Knowledge of differential and integral calculus in R is also required.
Subject matter
1. Topological notions in Rn: norms and metrics.
2. Functions of several variables: limits and continuity.
3. Differential calculus in Rn: partial derivatives; Schwarz theorem; differential of a function; directional derivatives; differentiability; Taylor''s Formula; implicit differentiation; inverse functions; maximum and minimum values; Lagrange Multipliers.
4. Multiple integrals: double and triple integrals; Fubini''s Theorem; change of variables in multiple integrals; applications of integrals; surface area; line integrals; the fundamental theorem for line integrals; Green''s Theorem; curl and divergence; parametric surfaces and their areas; surface integrals; Stokes''s Theorem; the divergence theorem.
5. Numerical series: geometric and telescopic series; convergence criteria for non negative series; absolute convergence; the Leibnitz criterion.
Bibliography
1- Cálculo vol. 2, Howard Anton, Irl Bivens, Stephen Davis,8ª edição,Bookman/Artmed
2- Calculus III, Jerrold Marsden and Alen Weinstein
3- Vector Calculus, Jerrold Marsden and Anthony Tromba, 5th edition
Teaching method
Theoretical classes consist on a theoretical exposition illustrated by examples of applications.
Practical classes consist on the solving of exercises of application of the methods and results presented in the theoretical classes.
Any questions or doubts will be adressed during the classes, during the weeekly sessions specially programmed to it or even at special sessions previously arranjed between professors and students.