Mathematical Analysis II


In this curricular unit, is intended that students develop their logical thinking ability and calculus, and learn the instruments that allow the formulation and resolution of problems within the scope of their study plan. The main goals are the acquisition and consolidation of the fundamental knowledge of differential and integral calculus in IR^n.

General characterization





Responsible teacher

Maria Helena da Costa Guerra Pereira


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 Evaluation Scheme (1st season):
Continuous assessment consists of conducting, during the academic semester, 2 tests in person, T 1, T 2 and  one final exam in person, E. The tests have no minimum grade. The exam has a minimum grade of 8.5 values.
The final grade is calculated as follows: 20% T 1+ 20% T2+ 60% E.
If, exceptionally, it is not possible to carry out an assessment in person, there will be the possibility of do the same online followed by an oral exam, but the situation will be analyzed case by case.

Examination regime (only 2nd season):
Final exam in person (100%)

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.