Mathematical Analysis II C


1- Understand the basic topological concepts in R  ( with special emphasis in R  R).

2. Understand the concepts of limit, continuity of real / vector functions of various real variables. Know how to calculate limits of the functions considered at indicated points.

3. Understand the concept of partial derivative, differentiability and its applications. Know how to calculate the derivative of the composite function and apply the implicit and inverse function theorems. Know how to determine the Taylor formula of a multiple variable function. Calculation of local, absolute and conditioned extremes.

4. Understand the concept of double and triple integral and know how to calculate these integrals using various types of coordinates.

5. Understand the concept of line and surface integrals and their applications: Green, Stokes and divergence theorems.

General characterization





Responsible teacher

João de Deus Mota Silva Marques


Weekly - 5

Total - 70

Teaching language



Students should know and apply the basic concepts of Mathematical Analysis I as well as linear algebra and analytic geometry.


1- Análise Matemática III-Cálculo diferencial em Rn . Vítor Faria e Silva. Ed. Danúbio, Lisboa 1983.

2- Análise matemática III-Cálculo diferencial em Rn (Fórmula de Taylor. Extremos). Vítor Faria e Silva. Ed. Danúbio, Lisboa 1983.

3- Cálculo vol. 2, Howard Anton, Irl Bivens, Stephen Davis,8ª edição,Bookman/Artmed

4- Calculus Vol.II, Apostol, T.M. , 1962.

5- Calculus III, Jerrold Marsden and Alen Weinstein.

6- Lições de análise infinitesimal.1- Cálculo diferencial em Rn . F.R. Dias Agudo, Lisboa 1969.

7- Vector Calculus, Jerrold Marsden and Anthony Tromba, 5ª edição

Teaching method

Theoretical classes consist of exposition of the subject, with the demonstration of the most relevant results, followed by illustrative examples of the subjects exposed.

In practical classes are solved exercises of application of the methods and results presented in the lectures. The exercises are preferably solved on the board by students accompanied by the clarification of the doubts that arise during their resolution. The exercises are chosen from a list previously made available in the Clip by the teachers.

There is a fixed time for clarifying doubts. There is also the possibility of clarifying doubts, beyond the time set for this purpose, in sessions previously agreed between teacher and students.

Evaluation method

Assessment Method - Mathematical Analysis II-C

In accordance with the Regulation of Knowledge Evaluation of the Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, revised on 30 July 2013, the discipline of Mathematical Analysis II-C has the following method of evaluation:



In the case of a repeating UC in its edition outside the curriculum, according to section six of article six of the RAC, the maximum number of unexcused absences fixed for a student to obtain attendance is four practical classes and six theoretical classes. Students with a status that exempts them are exempt from attendance.

Continuous evaluation

The continuous assessment of the discipline is carried out by performing three written tests (theoretical and practical evaluation) during the semester, each lasting not less than one hour and a half. Each written test will be assigned a rating (t1, t2 and t3) on the scale from 0 to 20 rounded to tenths.

The third test requires a minimum grade of 7.5 (t3≥7.5).

 The final grade of the Continuous Assessment "AC" is calculated by

(t1 + t2 + t3) / 3

to the units, by the usual conventions.

The student is approved for continuous assessment if he has obtained frequency according to the above rules if t3 ≥ 7.5 and if AC ≥10



Students who have been disqualified for continuous assessment with attendance or with a status that exempts them of it may take the exam.

The exam consists of a written test lasting no less than 3 hours than on the entire contents of the course.

The exam will be assigned an entire classification between 0 and 20 values, being the student approved to the discipline, with this classification, if it is greater than or equal to 10 values.

The use of calculating machines or any other calculation support instruments is prohibited at all times of evaluation.

Note Defense

Students with a final grade higher than or equal to 18 should complete a grade defense. Failure to complete this test leads to a final grade of 17 in the course. The final grade of a student who performs the grade defense will never be less than 17 points.

Rating Improvement

Students who are approved for continuous assessment may, through compliance with all the provisions imposed by FCT-UNL, apply for an improvement in classification. In that case, they may take the exam. The final classification will be the maximum between the marks obtained in continuous evaluation and in examination.

Conditions for carrying out the written tests (tests and examination)

Each of the Continuous Assessment and Examination tests may be submitted to students regularly enrolled in the Clip. Registration must be made until at least one week before the test (test or examination). On the day of the test each student must have a blank examination notebook that will be delivered at the beginning of the exam to the teacher who is carrying out the surveillance. During the test the student must carry an official identification document with a recent photograph.

In all that this Regulation is missing, the FCT-UNL General Regulations are valid.

Subject matter

1. Conics and Quadric surfaces (reviews)

2. Limits and Continuity in Rn

2.1. Topological notions in Rn

2.2. Vector valued functions and functions of several real variables: domain, graph, level curves and level surfaces.

2.3. Limits and continuity of  functions with several real variables.

3. Differential Calculus in Rn

3.1. Partial derivatives and Schwarz''''s theorem.

3.2. Directional derivative along a vector. Jacobian matrix, gradient vector and differentiability.

3.3. Differentiability of the composition of two functions. Taylor''''s theorem. Implicit and inverse function theorems.

3.4. Local extrema. Conditional extrema and Lagrange multipliers.

4. Integral Calculus in Rn

4.1. Double integrals. Iterated integrals and Fubini''''s theorem. Change of variable in double integrals. Double integrals in polar coodinates. Applications.

4.2. Triple integrals. Iterated integrals and Fubini''''s theorem. Change of variable in triple integrals. Triple integrals in cylindrical and spherical coordinates. Applications.

5. Vectorial Analysis

5.1. Vector fields: Gradient, divergence and curl. Closed fields. Gradient fields. Applications.

5.2. Formalism of differential forms. Line integrals of scalar and vector fields. Fundamental theorem of line integrals. Green''''s Theorem. Applications.

5.3. Surface integrals of scalar fields. Flux of a vector field across a surface. Stokes Theorem and Gauss-Ostrogradsky theorem. Applications.