# Mathematical Analysis II

## Objectives

Mathematical Analysis II is devoted to the study real of functions of several variables. Skills developed in
Mathematical Analysis 1 and Linear Algebra are put to test as a more comprehensive approach of analysis,
based on calculus and geometry, will treat fundamental issues such as optimization problems and Riemann
integration on planar or three dimensional domains.

## General characterization

100010

7.0

##### Responsible teacher

Mariana Iolanda Bárbara Dias

##### Hours

Weekly - Available soon

Total - Available soon

##### Teaching language

Portuguese. If there are Erasmus students, classes will be taught in English

### Prerequisites

Prerequisite recommended: Mathematical Analysis I

### Bibliography

Pires, C., Cálculo para Economia e Gestão, Escolar Editora, 2010.; Sydsæter, K, Hammond, P., Essential Mathematics for Economic Analysis, 2nd ed., Prentice Hall, 2006.; Sydsæter, K. et al., Further Mathematics for Economic Analysis, Prentice Hall, 2005.; Dias Agudo, F.R., Análise Real, Livraria Escolar Editora, 2ª edição, 1994.; Azenha, A., Jerónimo, M.A., Elementos de Cálculo Diferencial e Integral em ℝ e ℝn, McGraw-Hill, 1995.

### Teaching method

Lectures and pratical sessions with exercises.

### Evaluation method

Continuous Assessment System (only 1st season):
Final grade is calculate by the following formula: 3 intermediate Tests  (T1, T2, T3) during the semester (minimum grade in each test: 7,5 points)

Exam system (only 2nd season)
Final Exam (100%) (minimum grade: 9,5 points)

## Subject matter

1. Space IRn (n>=1)
Notion of norm and notion of distance;
Brief notions of topology.

2. Real functions of n real variables
General concepts and definitions.
Domain. Level Curves.
Limits and continuity.

3. Differential Calculus in IRn
Differentiability and differential.
Directional Derivative.
Higher-order derivatives and differential.
Derivative of the composite function.
Homogeneous function.
Taylor Formula.

4. Integral Calculus in IRn
The Riemann Integral.
Calculation of double integrals. Application to the calculation of areas.

5. Optimization
Some basic concepts.
Free optimization.
Equality constrained optimization: graphics resolution; method of Lagrange multipliers.
Inequality constrained optimization: graphic resolution.

## Programs

Programs where the course is taught: