# Mathematical Analysis II

**Chemical and Biochemical Engineering**

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

**Lecturer**

- Maria Cristina Oliveira da Costa

**Prerequisites**

Not applicable.

**Objectives**

1- To provide the mathematical foundations required in other modules of the programme.

2- 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: two written tests. Exam assessment: one written test.

**Bibliography**

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

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

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

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

**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**