Faculdade de Ciências e Tecnologia

Mathematical Analysis II D



Academic unit

Faculdade de Ciências e Tecnologia


Departamento de Matemática



Teacher in charge

Maria Fernanda de Almeida Cipriano Salvador Marques, Paula Cristiana Costa Garcia Silva Patrício

Weekly hours


Total hours


Teaching language



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


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- 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.
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. 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. Criterion of reason. D''Alembert criterion. Root criterion. Cauchy''s criterion.
6.3 Simple and absolute convergence. Alternate series and criterion of Leibnitz.


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.