Code: 23024
ECTS: 10
Departament: Department of Sciences and Technology
Field of Study: Mathematics
    Mathematical Analysis
    Nonstandard Analysis
A definir | To be defined

Course Description:
This unit contains an axiomatic introduction to Nonstandard Analysis. Its goals are the development of the calculus of infinitesimals and infinitely large numbers, and other orders of magnitude, as well as an introduction to basic notions of Nonstandard Analysis: S-continuity, S-differentiability, and S-integrability. 

Upon the conclusion of this LU the student should:
  • Capacity of formalizing and manipulating orders of magnitude of numbers.  
  • Insight in applicability and restrictions of the principle of mathematical induction and the principles of permanence.
  • Insight in the various forms of nonstandard regularity and irregularity, including infinitely fine discrete structures and constructions.  
  • Skills in asymptotic calculations and change of scale.  
  • Insight in the nature of problems of where nonstandard analysis is relevant, in particular problems with interplay of various orders of magnitude, problems with interactions between the discrete and the continuous and problems with imprecise transitions.

  1. Axiom of existence of nonstandard numbers; infinitesimal, limited and infinitely large numbers, Leibniz’s calculus rules. 
  2. Internal and external sets, principles of permanence.
  3. External induction.
  4. Analysis with infinitesimals, nonstandard notions of regularity of functions: S-continuity, S-derivability, S-integrability
  5. Orders of magnitude, change of scale.
  6. One from the special topics: singular perturbations, asymptotical approximations, infinitesimal discretizations.

  • Diener & Diener, Nonstandard analysis in practice, Springer, 1995;
  • Oliveira & van den Berg, Matemática Não Standard, Fundação Calouste Gulbenkian, 2007;
  • Nelson, Radically Elementary Probability Theory, Princeton, 1987


Workload (hours): 260
Contact Hours: 10

Evaluation is made on individual basis and it involves the coexistence of two modes: continuous assessment (60%) and final evaluation (40%). Further information is detailed in the Learning Agreement of the course unit.

This Learning Unit will not be open in 2015/16.

Language(s) of Instruction: Portuguese.

Contact for virtual mobility students: GDERI/International Affairs -