# Mathematical Analysis II

**Computer Engineering**, Publication in the Diário da República - Despacho n.º 8644/2020 - 08/09/2020

6 ECTS; 1º Ano, 2º Semestre, 70,0 TP

**Lecturer**

- Maria Cristina Oliveira da Costa

**Prerequisites**

Not applicable

**Objectives**

a) To provide the mathematical foundations required in other modules of the programme.

b) To provide the skills required to work with differential and integral calculus in functions of several real variables.

**Program**

CHAPTER I - Numerical and Function Series

Numerical series: definition and main properties.

Series of constant signal terms.

Absolute convergent and simply convergent series.

Operations with numeric series.

Function series.

Development of functions in power series.

Operations with development in power series

CHAPTER II - Real functions of several real variables

Introduction.

Limits and continuities.

Partial derivatives.

Differentiability.

Derivatives of composite functions.

Differentials of composite functions.

Derivation of implicitly defined functions.

Directional derivatives.

Homogeneous functions.

Local extremes.

Conditioned extremes.

CHAPTER III - Multiple Integrals

Double integrals:

Definition and properties.

Geometric interpretation of double integral as the volume of a solid

Double integrals in polar coordinates.

Applications of double integrals.

Triple integrals:

Definition and properties.

Triple integrals in cylindrical and spherical coordinates.

Applications of triple integrals.

**Evaluation Methodology**

The continuous assessment consists of three written tests. The first is rated from 0 to 6 values and following two tests are rated from 0 to 7 values. The student is passed by frequency if he obtains a grade of 10 or more, resulting from the sum of the three tests and at least 2 values on each test.

**Bibliography**

(2009). *Advanced Engineering Mathematics*. (Vol. 2). (pp. 1-1008). Sudbury: Jones & Bartlett Publishers

(1995). *Cálculo Diferencial e Integral em R e Rn*. (Vol. 1). (pp. 1-610). Lisboa: Mac Graw-Hill

(1999). *Princípios de Análise Matemática Aplicada*. (Vol. 1). (pp. 1-472). Lisboa: McGraw-Hill

(2012). *Introduction to calculus and analysis *. (Vol. II). New York: Springer Science & Business Media.

(2013). *Cálculo *. (Vol. II). São Paulo: São Paulo: Cengage Learning.

(1995). *Cálculo com Geometria Analítica*. (Vol. 1). (pp. 2-744). São Paulo: Makron Books

**Method of interaction**

Theoretical-practical lectures, with presentation and illustration of the proposed subjects and also exercises are proposed to be solved.

**Software used in class**

Not applicable