( english version )

Mathematical analisys AB ( 9 CFU ) Tel. 0521/032346 - Fax. 0521/032350 E-mail. giuseppe.mingione@unipr.it

Programma

The ¯eld of Real numbers R. Axioms for R: operations, order, completeness.

Maximum, minimum of a set. Relevant properties of sup and inf operators. The

triangle inequality.

The ¯eld of Complex numbers C. Complex numbers in the history of mathe-

matics: from the problem of solving algebraic equations to the abstract construc-

tion. Axioms for C. Algebraic and trigonometric form of complex numbers. Basic

properties and operations; miscellanea of manipulations. The triangle inequality in

C. The Gauss plane. Roots of zn = w and related problems. Geometric represen-

tation. The Fundamental Theorem of Algebra.

Limits. Elementary topology in R: intervals, neighborhoods and the fundamental

accumulation property. Functions and limits: basic properties. Operations with

limits: sum, products. Inde¯nite forms. A bit of sequences.

The functions exp and log. Qualitative behavior of the functions exp and log;

growth of functions and limits. Elementary algebraic properties. New inde¯nite

forms.

Continuous functions. Continuity of functions: basic properties. Right and

left limits. Zeros of a continuous function. Mean value theorem. Inversion of a

continuous function. Minima and maxima of a continuous function.

Derivatives. Di®erentiability of a function. Geometric interpretation, and tan-

gents to the graph. Di®erentiability and continuity. Derivatives and operations.

Basic use of derivatives: local maxima and minima. Checking increasing and de-

creasing functions via derivatives. Little Fermat's theorem. Classical theorems of

Rolle, Lagrange and Cauchy. How to use derivatives to compute limits: Taylor's

formula. Evaluation of error terms: at zero, at in¯nity.

Integration. How to invert the di®erentiation procedure. Basic integration meth-

ods: by parts, substitutions, and elementary algebraic tricks. The classical problem

of area evaluation: a sketch of the classical Peano construction. Basic properties and

theorems from integration theory; the integral as a linear operator. Non-integrable

functions. Classes of integrable functions. Integral mean value theorem and the

Fundamental Theorem of Calculus; its variations and consequences.

Generalized integrals. The problem of computing the area of unbounded sets.

Basic examples, comparison theorems. Standard convergence criteria.

Series. How to de¯ne the sum of in¯nite numbers. Basic examples, comparison

theorems. Standard convergence criteria. Series and generalized integrals.

Ordinary di®erential equations (ode). De¯nitions and physical motivations.

The Cauchy problem. Solvability of linear ode of ¯rst and second order.

Maximum, minimum of a set. Relevant properties of sup and inf operators. The

triangle inequality.

The ¯eld of Complex numbers C. Complex numbers in the history of mathe-

matics: from the problem of solving algebraic equations to the abstract construc-

tion. Axioms for C. Algebraic and trigonometric form of complex numbers. Basic

properties and operations; miscellanea of manipulations. The triangle inequality in

C. The Gauss plane. Roots of zn = w and related problems. Geometric represen-

tation. The Fundamental Theorem of Algebra.

Limits. Elementary topology in R: intervals, neighborhoods and the fundamental

accumulation property. Functions and limits: basic properties. Operations with

limits: sum, products. Inde¯nite forms. A bit of sequences.

The functions exp and log. Qualitative behavior of the functions exp and log;

growth of functions and limits. Elementary algebraic properties. New inde¯nite

forms.

Continuous functions. Continuity of functions: basic properties. Right and

left limits. Zeros of a continuous function. Mean value theorem. Inversion of a

continuous function. Minima and maxima of a continuous function.

Derivatives. Di®erentiability of a function. Geometric interpretation, and tan-

gents to the graph. Di®erentiability and continuity. Derivatives and operations.

Basic use of derivatives: local maxima and minima. Checking increasing and de-

creasing functions via derivatives. Little Fermat's theorem. Classical theorems of

Rolle, Lagrange and Cauchy. How to use derivatives to compute limits: Taylor's

formula. Evaluation of error terms: at zero, at in¯nity.

Integration. How to invert the di®erentiation procedure. Basic integration meth-

ods: by parts, substitutions, and elementary algebraic tricks. The classical problem

of area evaluation: a sketch of the classical Peano construction. Basic properties and

theorems from integration theory; the integral as a linear operator. Non-integrable

functions. Classes of integrable functions. Integral mean value theorem and the

Fundamental Theorem of Calculus; its variations and consequences.

Generalized integrals. The problem of computing the area of unbounded sets.

Basic examples, comparison theorems. Standard convergence criteria.

Series. How to de¯ne the sum of in¯nite numbers. Basic examples, comparison

theorems. Standard convergence criteria. Series and generalized integrals.

Ordinary di®erential equations (ode). De¯nitions and physical motivations.

The Cauchy problem. Solvability of linear ode of ¯rst and second order.