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

Paula Cristiana Costa Garcia Silva Patrício


Weekly - 6

Total - 28

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


1.You can only submit to any of the written tests and consequently be evaluated in this course unit, students who: a) have registered to take the test through the Clip page, up to a week before the respective date, and delivering a blank ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''examination notebook'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' (''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''caderno de teste'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''') on the day of the test; b). in the act of proof, bear an official identification document, stating a photograph (for example, Citizen''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s Card, Identity Card, Passport, some versions of Student Card).

2. It is not allowed to use any calculator in evaluation tests or exams. It is not advisable to use a graphing calculator in class.

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.

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.


Programs where the course is taught: