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Subject: FONDAMENTI DI ANALISI MATEMATICA (A.A. 2021/2022)

master degree course in MATHEMATICS

Course year 2
CFU 6
Teaching units Unit Fondamenti di analisi matematica
Related or Additional Studies (lesson)
  • TAF: Supplementary compulsory subjects SSD: MAT/04 CFU: 6
Teachers: Arrigo BONISOLI, Carlo 6/8/1962 BENASSI
Exam type oral
Evaluation final vote
Teaching language Italiano
Contents download pdf download

Teachers

Carlo 6/8/1962 BENASSI
Arrigo BONISOLI

Overview

Module: ANALISI MATEMATICA PER L'INSEGNAMENTO (LM in Matematica)/ Course: FONDAMENTI DI MATEMATICA (LM in Didattica e Comunicazione delle Scienze)

The course analyse the fundamental steps that brought to the conept of derivative, starting from the problem of calculus of tangent to a curve in the ancient Greece till the fundamental contribution due to Lagrange. The course provide hints for a fruitful teaching of the differential calculus in one variable.

Module: FONDAMENTI DI ANALISI MATEMATICA
The course gives an overview of the fundamental steps that originated the modern theory of integration and the theory of sets. It provides hints for a fruitful teaching of the theory of integration and a rigorous approach to the theory of sets.

Admission requirements

Module: ANALISI MATEMATICA PER L'INSEGNAMENTO (LM in Matematica)/ Course: FONDAMENTI DI MATEMATICA (LM in Didattica e Comunicazione delle Scienze)

Theory of differential calculus, in the one dimensional case.

Module: FONDAMENTI DI ANALISI MATEMATICA
Riemann's theory of integration. The language of elementary set theory,

Course contents

Modulo: ANALISI MATEMATICA PER L'INSEGNAMENTO
The problem of calculus of tangents to a curve in the ancient Greece: the contribution of Euclide and Apollonio.
Cartesio: analytical geometry and tangents
Fermat: adequalities and first derivative
Cinematic construction of tangents
Newton and its method of fluxions
Application to fluxion calculus to problems of maximum and minimim
Method of fluxions to solve the problem of tangent to a curve
From fluxions to fluents: quadrature of a curve
Leibniz: the differential calculus
Leibniz: tangents, maxima and minima
Diffusion of infinitesimal calculus
An introduction to the Calculus of Variations
The isoperimetric problem
Maximum and minimum problems
The diffusion of infinitesimal calculus: from Bernoulli's brothers to Lagrange

Module: FONDAMENTI DI ANALISI MATEMATICA
The evolution of the concept of a function in the nineteenth century,
Cauchy's integral and Riemann's reformulation.
Vitali's contribution to the development of the theory of integration.
The origins of the concept of a measure. Baire's theory and Henstock's integral. The contribution of Cantosr: the birth of the theory of sets.
Cardinal numbers. The axiom of choice in its various formulations.

Teaching methods

Module: ANALISI MATEMATICA PER L'INSEGNAMENTO Frontal lectures with slides Module: FONDAMENTI DI ANALISI MATEMATICA Frontal lectures

Assessment methods

Module: ANALISI MATEMATICA PER L'INSEGNAMENTO (LM in Matematica)/ Course: FONDAMENTI DI MATEMATICA (LM in Didattica e Comunicazione delle Scienze) An oral interview on the course material, with possibly in addiction a seminar on some topics of further study Module: FONDAMENTI DI ANALISI MATEMATICA An oral interview on the course material

Learning outcomes

1. Knowledge and understanding:
at the end of the course a student will have a deep knowledge of the theoretical problems that yielded the modern theory of differential calculus, integration and the theory of sets.

2. Applying knowledge and understanding:
a deep knowledge of the historical evolution of the concept of derivative, integral and of the theory of sets will allow the student to set forth autonomous learning paths and strategies in order to efficiently teach the theory of differential calculus, integration and the theory of sets.

3. Making judgements:
at the end of the course a student should have improved his/her capability of handling theoretical arguments and recognize their formal correctness.

4. Communicating skills:
at the end of the course a student should be able to describe topics in the theory of differential calculus, integration and in the theory of sets with an appropriate technical language and a correct mathematical formalism

5. Learning skills:
studying the subject matter (occasionally on textbooks in English) should stimulate independent learning skills and the capability of treating connected topics in further detail

Readings

Modulo: ANALISI MATEMATICA PER L'INSEGNAMENTO (LM in Matematica)/ Corso: FONDAMENTI DI MATEMATICA (LM in Didattica e Comunicazione delle Scienze)

Slide del corso e bibliografia consigliata

Modulo: FONDAMENTI DI ANALISI MATEMATICA

T. Hawkins
Lebesgue's theory of integration: its origins and development
AMS

L. Lombardo Radice,
Istituzioni di algebra astratta,
Feltrinelli, Milano, 1965

P.R. Halmos,
Naive Set Theory.
D. Van Nostrand Company, Princeton, NJ, 1960.
Reprinted by Springer-Verlag, New York, 1974.
ISBN 0-387-90092-6 (Springer-Verlag edition).
Reprinted by Martino Fine Books, 2011.
ISBN 978-1-61427-131-4 (Paperback edition).

Traduzione italiana:
Teoria elementare degli insiemi,
a cura di di M.L. Vesentini Ottolenghi, E. Vesentini,
Feltrinelli, Milano 1982.
ISBN-10: 8807620073. ISBN-13: 9788807620072.