Mathematical Analysis II C
Code
10347
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Maria Fernanda de Almeida Cipriano Salvador Marques
Weekly hours
5
Total hours
70
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 mathematical analysis of functions of one variable corresponding to the completion of the course of Mathematical Analysis IIC. 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, the matrix representation of linear functions defined on R ^ n with values on R ^ m and matrix calculation.
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
Teaching method
The professor gives the course by lectures, where he explains all topics referred to in the syllabus. Problem sheets are provided to students to be worked outside the classroom with prior knowledge acquired during the course. Practical classes are taught, where the teacher clarifies the doubts about the problems given previously and the more relevant problems are solved in the blackboard.
Students still have the so-called "horário de dúvidas" where they can clarify their doubts with the teacher
Evaluation method
Frequency is given to students who :
a) attend at least two thirds of classes taught ; and b ) deliver ( the teacher who teaches the practical part ) the resolutions of all the exercises proposed for obtaining frequency within the specified period.
Lists of exercises proposed for obtaining frequency as well as the delivery dates of the resolutions are published in the clip during the semester .
Students must have frequency to realize the evaluations. Students with student worker status are exempt frequency .
The evaluation consists of three mid-term tests, or of a final exam. The three tests can replace the final exam, in case of approval. More detailed rules are available in the Portuguese version.