# Mathematical Analysis II

**Chemical and Biochemical Engineering**, Publication in the Diário da República - Despacho nº 10764/2011 - 30/08/2011

6 ECTS; 1º Ano, 2º Semestre, 30,0 T + 30,0 TP

**Lecturer**

**Prerequisites**

Not applicable.

**Objectives**

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

b) To provide the students with skills to work with differential and integral calculus of 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.

Some applications of double integrals.

Triple integrals:

Definition and properties.

Triple integrals in cylindrical and spherical coordinates.

Some applications of triple integrals.

**Evaluation Methodology**

Continuous assessment consists of three written tests. The first one is graded from 0 to 6 and the following two from 0 to 7. A minimum mark of 2 in each test and a minimum overall mark of 10 exempts students from taking the exam.

**Bibliography**

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

- Zill, D. e Cullen, M. (2009). *Advanced Engineering Mathematics*. (Vol. 1). (pp. 1-1008). Sudbury: Jones & Bartlett Publishers

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

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

- Stewart, J. (2013). *Cálculo *. (Vol. II). São Paulo: São Paulo

- Courant, . e John, F. (2012). *Introduction to calculus and analysis*. (Vol. II). New York: Springer Science & Business Media.

**Method of interaction**

Theoretical lectures, with presentation and illustration of the proposed subjects. Theoretical-practical lectures in which exercises are proposed and solved.

**Software used in class**

Not applicable.