Mathematical Analysis II


This course is intended for students to develop their capacity for logical reasoning and calculation, and learn the tools for formulating and solving problems posed in its syllabus. The main purposes are the acquisition and consolidation of fundamental knowledge of differential and integral calculus in IRn.

General characterization





Responsible teacher

Patrícia Santos Ribeiro


Weekly - Available soon

Total - Available soon

Teaching language

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


Prerequisite recommended: Mathematical Analysis I


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 IR e IRn, 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)
Final Grade: 30%T1+40%T2+30%T3 
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
Partial Derivatives. Gradient.
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.