Mathematical Analysis II D


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

General characterization





Responsible teacher

José Maria Nunes de Almeida Gonçalves Gomes, Paula Cristiana Costa Garcia Silva Patrício


Weekly - 4

Total - 66

Teaching language



The students should have knowledge of mathematical analysis of functions of one variable corresponding to the completion of the course of Mathematical Analysis I. 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.


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

Continuous evaluation.

In order to be eligible for continuous evaluation, the student may attend 2/3 of pratical sessions or 2/3 of theoretical sessions. The continuous evaluation method is based upon two tests. The student is aproved if the grade average is greater  or equal to 9.5. In the final exam date, the student may optionally improve one of the tests. If an approved student desires to improve the grade by exam he   must express his intention to the responsible teacher so that the aproval grade is previously registered.


Exam evaluation

Any student inscribed in the discpline may attend to the exam, being aaproved if the final grade is greater or equal than 9.5.


If necessary, the responsible teacher  may individually require an extraordinary evaluation.

Subject matter

1- Review of some concepts of Analytical Geometry
1.1 Conics.
1.2 Quadric.
2- Limits and continuity in R ^ n
2.1 Topological notions in R ^ n.
2.2 Vector functions and functions of several real variables: domain, graph,  level curves and level surfaces.
2.3 Limit and continuity of functions of several real variables
3- Differential calculus in R ^ n
3.1 Partial derivatives and Schwarz''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem.
3.2 Directional derivative. Jacobian matrix, vector gradient and notion of differentiability.
3.3 Differentiability of the composite function. Taylor''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem. Implicit function theorem and inverse function theorem.
3.4 Relative extremes. Lagrange conditioners and multipliers.
4- Integral calculation in R ^ n
4.1 Double integrals. Iterated integrals and Fubini''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem. Change of variable in double integrals. Double integrals in polar coordinates. Applications.
4.2 Triple integrals. Iterated integrals and Fubini''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem. Variable change in triple integrals. Triple integrals in cylindrical and spherical coordinates. Applications.
5- Vector analysis
5.1 Vector fields: gradient, divergence and rotational. closed Fields. Conservative fields. Applications.
5.2 Formalism of differential forms. Line integrals of scalar fields and vector fields. Fundamental theorem for line integrals. Theorem of Green. Applications.
5.3 Surface integrals of scalar fields. Flow of a vector field across a surface. Stokes''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem and Gauss-Ostrogradsky''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem. Applications.
6- Numerical series
6.1 Convergence of numerical series. Necessary condition of convergence. Telescopic series. Geometric series.
6.2 Series of non-negative terms. Dirichlet series. Criteria for comparison. Ratio criterion. D''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''Alembert criterion. Root criterion. Cauchy''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s criterion.
6.3 Simple and absolute convergence. Alternate series and criterion of Leibnitz.