To acquire a good knowledge of the basic concepts and methods of Mathematical Analysis with particular regard to the structure of real numbers, to the theory of limits, to the study of functions and to the first applications and models.
Curriculum
teacher profile teaching materials
• Sets, relations and functions.
• Axioms of real numbers.
• Elementary properties of ordered fields.
• Symmetric sets and functions. Absolute value and distance.
• Natural numbers. Subtraction in N; principle of well-ordering and its consequences.
• Sequences and recursion theorem (optional proof). Recursive definition of sums, products and powers.
• N^th powers, geometric sum and formula for a^n-b^n. Newton's binomial formula.
• Finite and infinite sets.
• Rational numbers. The rationals are countable. Gauss lemma.
• Least upper bound and greatest lower bound. Elementary consequences of the completeness axiom on integers.
• Roots. Powers with rational exponent.
• Monotone functions.
PART 2: Theory of limits
• The extended real system R*. Intervals and neighbourhoods.
• Internal, isolated, accumulation points. General definition of limit. Uniqueness of the limit.
• Sign permanence theorem. Comparison theorems.
• Side limits and monotone functions.
• Algebra of finite limits. Extended limit algebra.
• Some notable limits of sequences.
• The number of Nepero.
• Bridge theorem and characterisation of the sup / inf by sequences.
• Continuity: general considerations; theorem of existence of zeros. Intermediate value theorem.
• Classification of discontinuities.
• Limits for compound functions.
• Limits for inverse functions.
• A continuous and strictly monotone function on an interval admits a continuous inverse.
• Logarithms.
• Notable limits (exponential and logarithms).
PART 3: Series
• Numerical series: Elementary properties of series. Comparison criteria.
• Decimal expansions.
• Convergence criteria for series with positive terms.
• Criteria for series with real terms (Abel-Dirichlet, Leibniz).
• Exponential series. Irrationality of e. Speed of divergence of the harmonic series.
• Properties of trigonometric functions (in particular proof of the cosine addition theorem).
• Periodic functions. Monotonic properties of trigonometric functions.
• Inverse trigonometric functions.
McGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Exercise texts:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Programme
PART 1: The set of real numbers and its main subsets• Sets, relations and functions.
• Axioms of real numbers.
• Elementary properties of ordered fields.
• Symmetric sets and functions. Absolute value and distance.
• Natural numbers. Subtraction in N; principle of well-ordering and its consequences.
• Sequences and recursion theorem (optional proof). Recursive definition of sums, products and powers.
• N^th powers, geometric sum and formula for a^n-b^n. Newton's binomial formula.
• Finite and infinite sets.
• Rational numbers. The rationals are countable. Gauss lemma.
• Least upper bound and greatest lower bound. Elementary consequences of the completeness axiom on integers.
• Roots. Powers with rational exponent.
• Monotone functions.
PART 2: Theory of limits
• The extended real system R*. Intervals and neighbourhoods.
• Internal, isolated, accumulation points. General definition of limit. Uniqueness of the limit.
• Sign permanence theorem. Comparison theorems.
• Side limits and monotone functions.
• Algebra of finite limits. Extended limit algebra.
• Some notable limits of sequences.
• The number of Nepero.
• Bridge theorem and characterisation of the sup / inf by sequences.
• Continuity: general considerations; theorem of existence of zeros. Intermediate value theorem.
• Classification of discontinuities.
• Limits for compound functions.
• Limits for inverse functions.
• A continuous and strictly monotone function on an interval admits a continuous inverse.
• Logarithms.
• Notable limits (exponential and logarithms).
PART 3: Series
• Numerical series: Elementary properties of series. Comparison criteria.
• Decimal expansions.
• Convergence criteria for series with positive terms.
• Criteria for series with real terms (Abel-Dirichlet, Leibniz).
• Exponential series. Irrationality of e. Speed of divergence of the harmonic series.
• Properties of trigonometric functions (in particular proof of the cosine addition theorem).
• Periodic functions. Monotonic properties of trigonometric functions.
• Inverse trigonometric functions.
Core Documentation
Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su RMcGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Exercise texts:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Reference Bibliography
John M.H. Olmsted The Real Number System , 1962, 216 pages Giusti, E.: Analisi Matematica 1, Seconda Edizione Bollati Boringhieri, 1991 (edizione fuori commercio; Libreria Efesto ) Giusti, E.: Analisi Matematica 1, Terza Edizione Bollati Boringhieri, 2002 Rudin, W.: Principi di analisi matematica, Milano 1991 (edizione fuori commercio) Bertsch, Dal Passo, Giacomelli - Analisi Matematica - McGraw-Hill (2011) - piattaforma Connect (esercizi a scelta multipla)Type of delivery of the course
Lectures (about forty-eight hours) and exercises (about forty-two hours). All the program material will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself.Attendance
Attendance is optional and understanding of the text adopted is sufficient for full use of the course. Of course attendance is desirable and strongly recommended as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.Type of evaluation
The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher. teacher profile teaching materials
• Sets, relations and functions.
• Axioms of real numbers.
• Elementary properties of ordered fields.
• Symmetric sets and functions. Absolute value and distance.
• Natural numbers. Subtraction in N; principle of well-ordering and its consequences.
• Sequences and recursion theorem (optional proof). Recursive definition of sums, products and powers.
• N^th powers, geometric sum and formula for a^n-b^n. Newton's binomial formula.
• Finite and infinite sets.
• Rational numbers. The rationals are countable. Gauss lemma.
• Least upper bound and greatest lower bound. Elementary consequences of the completeness axiom on integers.
• Roots. Powers with rational exponent.
• Monotone functions.
PART 2: Theory of limits
• The extended real system R*. Intervals and neighbourhoods.
• Internal, isolated, accumulation points. General definition of limit. Uniqueness of the limit.
• Sign permanence theorem. Comparison theorems.
• Side limits and monotone functions.
• Algebra of finite limits. Extended limit algebra.
• Some notable limits of sequences.
• The number of Nepero.
• Bridge theorem and characterisation of the sup / inf by sequences.
• Continuity: general considerations; theorem of existence of zeros. Intermediate value theorem.
• Classification of discontinuities.
• Limits for compound functions.
• Limits for inverse functions.
• A continuous and strictly monotone function on an interval admits a continuous inverse.
• Logarithms.
• Notable limits (exponential and logarithms).
PART 3: Series
• Numerical series: Elementary properties of series. Comparison criteria.
• Decimal expansions.
• Convergence criteria for series with positive terms.
• Criteria for series with real terms (Abel-Dirichlet, Leibniz).
• Exponential series. Irrationality of e. Speed of divergence of the harmonic series.
• Properties of trigonometric functions (in particular proof of the cosine addition theorem).
• Periodic functions. Monotonic properties of trigonometric functions.
• Inverse trigonometric functions.
McGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Exercise texts:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Programme
PART 1: The set of real numbers and its main subsets• Sets, relations and functions.
• Axioms of real numbers.
• Elementary properties of ordered fields.
• Symmetric sets and functions. Absolute value and distance.
• Natural numbers. Subtraction in N; principle of well-ordering and its consequences.
• Sequences and recursion theorem (optional proof). Recursive definition of sums, products and powers.
• N^th powers, geometric sum and formula for a^n-b^n. Newton's binomial formula.
• Finite and infinite sets.
• Rational numbers. The rationals are countable. Gauss lemma.
• Least upper bound and greatest lower bound. Elementary consequences of the completeness axiom on integers.
• Roots. Powers with rational exponent.
• Monotone functions.
PART 2: Theory of limits
• The extended real system R*. Intervals and neighbourhoods.
• Internal, isolated, accumulation points. General definition of limit. Uniqueness of the limit.
• Sign permanence theorem. Comparison theorems.
• Side limits and monotone functions.
• Algebra of finite limits. Extended limit algebra.
• Some notable limits of sequences.
• The number of Nepero.
• Bridge theorem and characterisation of the sup / inf by sequences.
• Continuity: general considerations; theorem of existence of zeros. Intermediate value theorem.
• Classification of discontinuities.
• Limits for compound functions.
• Limits for inverse functions.
• A continuous and strictly monotone function on an interval admits a continuous inverse.
• Logarithms.
• Notable limits (exponential and logarithms).
PART 3: Series
• Numerical series: Elementary properties of series. Comparison criteria.
• Decimal expansions.
• Convergence criteria for series with positive terms.
• Criteria for series with real terms (Abel-Dirichlet, Leibniz).
• Exponential series. Irrationality of e. Speed of divergence of the harmonic series.
• Properties of trigonometric functions (in particular proof of the cosine addition theorem).
• Periodic functions. Monotonic properties of trigonometric functions.
• Inverse trigonometric functions.
Core Documentation
Luigi Chierchia: Corso di analisi. Prima parte. Una introduzione rigorosa all'analisi matematica su RMcGraw-Hill Education Collana: Collana di istruzione scientifica
Data di Pubblicazione: giugno 2019
EAN: 9788838695438 ISBN: 8838695431
Pagine: XI-374 Formato: brossura
https://www.mheducation.it/9788838695438-italy-corso-di-analisi-prima-parte
Exercise texts:
Giusti, E.: Esercizi e complementi di Analisi Matematica, Volume Primo, Bollati Boringhieri, 2000
Demidovich, B.P., Esercizi e problemi di Analisi Matematica, Editori Riuniti, 2010
Reference Bibliography
John M.H. Olmsted The Real Number System , 1962, 216 pages Giusti, E.: Analisi Matematica 1, Seconda Edizione Bollati Boringhieri, 1991 (edizione fuori commercio; Libreria Efesto ) Giusti, E.: Analisi Matematica 1, Terza Edizione Bollati Boringhieri, 2002 Rudin, W.: Principi di analisi matematica, Milano 1991 (edizione fuori commercio) Bertsch, Dal Passo, Giacomelli - Analisi Matematica - McGraw-Hill (2011) - piattaforma Connect (esercizi a scelta multipla)Type of delivery of the course
Lectures (about forty-eight hours) and exercises (about forty-two hours). All the program material will be explained in class. The lessons / exercises will include a continuous dialogue with the students: the feedback from the students during the course is a fundamental tool for the success of the course itself.Attendance
Attendance is optional and understanding of the text adopted is sufficient for full use of the course. Of course attendance is desirable and strongly recommended as the interaction between teacher and students is a fundamental and unrepeatable teaching tool.Type of evaluation
The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher.