Mathematical Analysis II

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
1st Cycle Studies

Recommended Prerequisites

Mathematical Analysis I and Linear Algebra and Analytic Geometry.   

Teaching Methods

The teaching in this course assumes two formats: theoretical and example classes. During a theoretical class teaching is mostly expository. During an example class teaching consists of problem solving by the students under the guidance of the lecturer. A strong interaction between notions and their practical application is emphasized. In this task, the visualization and the analysis of concrete examples takes on a central role and prepares the way for the abstract definitions. Tutorial support is available to students to help them on the tasks assigned by the lecturers.

Learning Outcomes

The student who successfully completes this course will be able to:

1. Detect non-continuous real functions of two variables at a given point;

2. Compute the directions of greatest growth of a real function of two variables;

3. Solve a constrained extrema problem;           

4. Compute areas and volumes, as well as the center of mass of a solid (with arbitrary density function), using double and triple integrals;

5. Compute areas and lengths of curves in space using line and surface integrals;            

6. Solve problems involving the connections between the types of integrals studied (double, triple, line and surface integrals);

7. Solve problems involving applications of integration to mathematical modelling.    

Work Placement(s)



I. Real functions of several variables

I.1 Limits and continuity

I.2 Partial derivatives (gradient, tangent plane, differentiability)

I.3 Directional derivative

I.4 Chain Rule

I.5 Vector valued functions. Jacobian matrix

I.6 Extrema of functions. Lagrange multipliers


II Integral calculus in R2 and R3

II.1 Double integrals and applications

II.2 Triple integrals and applications

II.3 Change of variables (polar, cylindrical, and spherical coordinates)

II.4 Line integrals. Green's Theorem

II.5 Surface integrals. Stoke's and divergence theorems.    

Head Lecturer(s)

Susana Margarida Pereira da Silva Domingues de Moura

Assessment Methods

Continuous assessment
2 or more midterm exams: 100.0%

Final assessment
Exam: 100.0%


[1] James Stewart: Cálculo, Volumes I e II, Cengage Learning (tradução da 8ª edição norte-americana) 2017

 [2] Gabriel E. Pires: Cálculo diferencial e integral em Rn. IST Press (Colecção Ensino da Ciência e da Tecnologia), 2012.

 [3] Ana d'Azevedo Breda, Joana Nunes da Costa: Cálculo com funções de várias variáveis. McGraw-Hill, Lisboa (1996).

 [4] M. Olga Baptista: Matemática - Integrais Duplos, Triplos, de Linha e de Superfície. Edições Sílabo. (2ª Edição: 2001).