# 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**

- 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 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**

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.

Exam assessment: Written test classified from 0 to 20 values. The student is approved if he obtains a classification higher or equal to 10 values.

For any of the evaluations, if the student obtains a classification of 15 or higher, he should be subject to an extraordinary assessment.

**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.