Mathematical Analysis II D


The aproved student should be abble to study regularity of functions of several variables in their domains, determine maximums and minimums values by identifying their extremals and to compute line, area, surface and volume integrals.  

Also de student should know and apply the classical results on the convergence of numerical series.

General characterization





Responsible teacher

Ana Maria de Sousa Alves de Sá, José Maria Nunes de Almeida Gonçalves Gomes


Weekly - 4

Total - 56

Teaching language



The student should have the required skills of aproval to basic Analytical Geometry, Analysis 1 and Linear Algebra.


Calculus; Anton, Bivens and Davis, Wiley (8th edition)

Curso de Análise, vol 2; Elon L. Lima, Ed IMPA (projecto Euclides)

Cálculo, vol 2; Tom M. Apostol, Ed. Reverté.

Principles of Mathematical Analysis, W. Rudin, McGraw Hill.

Teaching method

Our teaching methos is based upon a dual approach, theoretical and pratical. In the theoretical sessions, concepts are introduced providing their motivation as well as its applications in the context of engineering. Problem solving sessions will allow the revision of the fundamental theoretical notions, as well as develop student''s operational skills in view of their utility in a professional context.


Besides bibliography, students will have the support of a theoretical-pratical text (available on CLIP) as well as of an attending schedule where individual questions and dificulties can be treated.

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-      Analytical Geometry (revisions).

1.1   Conics.

1.2   Quadrics.

2-      Limits and continuity in Rn.

2.1   Topological notions in Rn.

2.2   Vector functions of a real variable and real functions of several variables: domain, graphic, level curves and surfaces.

2.3  Limits and continuity.

3-    Differential calculus in Rn

3.1   Partial derivatives. Schwarz''s Theorem.

3.2   Derivative along a vector. Jacobian matrix. Gradient. Differentiability.

3.3   Differentiability of a composition of functions. Taylor''s Theorem. Implicit function Theorem. Invers Function Theorem.

3.4   Extremums. Lagrange Multipliers Theorem.


4-      Integral calculus in Rn

4.1   Double integrals. Fubini''s Theorem. Change of variables theorem for double integrals.  Polar coordinates. Applications.

4.2    Triple integrals. Fubini''s Theorem. Change of variables in triple integrals. Cylindrical ans spherical coordinates. Applications.


5-      Vectorial analysis

5.1   Vector fields. Gradient Field. Divergence and rotational. Closed fields. Conservative fields. Applications.

5.2   Differential forms. Line integrals of scalar and vector fields. Fundamental Theorem for line integrals. Green''s Theorem. Applications.

5.3   Surface integrals for scalar fields. Vector flow on a surface. Stokes Theorem and Gauss-Ostrogradsky Theorem. Applications.

6-      Numerical series

6.1   Convergence of a numerical series. Necessary condition for convergence. Telescopic series. Geometrical series.

6.2   Series of non-negative terms. Dirichlet series. Convergence criteria.

6.3   Absolute and simple convergence. Leibnitz criteria for alternate series.