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Mathematical analysis 1

A.Y. 2016/2017

Learning objectives

Il corso si propone di fornire allo studente un'introduzione e un primo approfondimento della conoscenza dell'Analisi Matematica con particolare riferimento ai numeri reali, numeri complessi, successioni e serie numeriche , limiti,

continuita', calcolo differenziale in una variabile. Le nozioni di

limite e continuita' sono trattate nell'ambito piu' astratto degli spazi

metrici, di cui viene fornita una trattazione semplice ma precisa.

continuita', calcolo differenziale in una variabile. Le nozioni di

limite e continuita' sono trattate nell'ambito piu' astratto degli spazi

metrici, di cui viene fornita una trattazione semplice ma precisa.

Expected learning outcomes

Undefined

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

First semester

**Course syllabus**

Main topics of the course, a.y. 2011/12

The real and complex number systems

The set of real numbers: an ordered field with the least upper bound property. Existence of the n-th root of a positive real number. Decimal expansions. The extended real number system. The complex field: definition and its main properties. Algebraic, trigonometric and exponential forms of a complex number. Operations with complex numbers. De Moivre's formula. The n-th roots of a complex number. The Fundamental Theorem of Algebra and its consequences.

Sets, functions and metric spaces

Basic topics on sets and functions. Equipotent sets. Finite, countable and uncountable sets. Uncountability of R. The normed vector space R^n. Cauchy-Schwartz' inequality. Metric spaces and their topology: bounded, open, closed, compact and connected sets. Compactification of R.

Sequences

Properties of convergent sequences in a metric space. Cauchy sequences. Subsequences. Sequences of real numbers. Limits and their operations. Monotone sequences. The number e. Special limits.

Numerical series

Convergence and divergence of numerical series. Absolute convergence. Cauchy's criterion for convergence. Sufficient criteria for absolute convergence. Alternating series and the Leibnitz criterion.

Mappings between metric spaces

Limits of functions: metric and sequential definitions. Pointwise and global continuity. Inverse images of open sets Continuity, compactness, connectedness. Continuity, composition and invertibility. Uniform continuity. Real functions of one real variable. Limits for monotonic functions. Asymptotic behaviour. Discontinuities.

Differential calculus for real functions of one real variable

Differentiability: geometrical meaning and continuity. Differentiation rules. The derivatives of elementary functions. Higher derivatives. Differentiability, composition, invertibility. The theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. L'Hospital's theorems. Taylor's formula with the Peano and Lagrange forms for the remainder. Mac-Laurin's formula for elementary functions. Maxima and minima. Convexity in an interval. Inflection points.

The real and complex number systems

The set of real numbers: an ordered field with the least upper bound property. Existence of the n-th root of a positive real number. Decimal expansions. The extended real number system. The complex field: definition and its main properties. Algebraic, trigonometric and exponential forms of a complex number. Operations with complex numbers. De Moivre's formula. The n-th roots of a complex number. The Fundamental Theorem of Algebra and its consequences.

Sets, functions and metric spaces

Basic topics on sets and functions. Equipotent sets. Finite, countable and uncountable sets. Uncountability of R. The normed vector space R^n. Cauchy-Schwartz' inequality. Metric spaces and their topology: bounded, open, closed, compact and connected sets. Compactification of R.

Sequences

Properties of convergent sequences in a metric space. Cauchy sequences. Subsequences. Sequences of real numbers. Limits and their operations. Monotone sequences. The number e. Special limits.

Numerical series

Convergence and divergence of numerical series. Absolute convergence. Cauchy's criterion for convergence. Sufficient criteria for absolute convergence. Alternating series and the Leibnitz criterion.

Mappings between metric spaces

Limits of functions: metric and sequential definitions. Pointwise and global continuity. Inverse images of open sets Continuity, compactness, connectedness. Continuity, composition and invertibility. Uniform continuity. Real functions of one real variable. Limits for monotonic functions. Asymptotic behaviour. Discontinuities.

Differential calculus for real functions of one real variable

Differentiability: geometrical meaning and continuity. Differentiation rules. The derivatives of elementary functions. Higher derivatives. Differentiability, composition, invertibility. The theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. L'Hospital's theorems. Taylor's formula with the Peano and Lagrange forms for the remainder. Mac-Laurin's formula for elementary functions. Maxima and minima. Convexity in an interval. Inflection points.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8

Practicals: 40 hours

Lessons: 32 hours

Lessons: 32 hours

Professors:
De Bernardi Carlo Alberto, Zanco Clemente

### CORSO B

Responsible

Lesson period

First semester

**Course syllabus**

Main topics of the course, a.y. 2011/12

The real and complex number systems

The set of real numbers: an ordered field with the least upper bound property. Existence of the n-th root of a positive real number. Decimal expansions. The extended real number system. The complex field: definition and its main properties. Algebraic, trigonometric and exponential forms of a complex number. Operations with complex numbers. De Moivre's formula. The n-th roots of a complex number. The Fundamental Theorem of Algebra and its consequences.

Sets, functions and metric spaces

Basic topics on sets and functions. Equipotent sets. Finite, countable and uncountable sets. Uncountability of R. The normed vector space R^n. Cauchy-Schwartz' inequality. Metric spaces and their topology: bounded, open, closed, compact and connected sets. Compactification of R.

Sequences

Properties of convergent sequences in a metric space. Cauchy sequences. Subsequences. Sequences of real numbers. Limits and their operations. Monotone sequences. The number e. Special limits.

Numerical series

Convergence and divergence of numerical series. Absolute convergence. Cauchy's criterion for convergence. Sufficient criteria for absolute convergence. Alternating series and the Leibnitz criterion.

Mappings between metric spaces

Limits of functions: metric and sequential definitions. Pointwise and global continuity. Inverse images of open sets Continuity, compactness, connectedness. Continuity, composition and invertibility. Uniform continuity. Real functions of one real variable. Limits for monotonic functions. Asymptotic behaviour. Discontinuities.

Differential calculus for real functions of one real variable

Differentiability: geometrical meaning and continuity. Differentiation rules. The derivatives of elementary functions. Higher derivatives. Differentiability, composition, invertibility. The theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. L'Hospital's theorems. Taylor's formula with the Peano and Lagrange forms for the remainder. Mac-Laurin's formula for elementary functions. Maxima and minima. Convexity in an interval. Inflection points.

The real and complex number systems

The set of real numbers: an ordered field with the least upper bound property. Existence of the n-th root of a positive real number. Decimal expansions. The extended real number system. The complex field: definition and its main properties. Algebraic, trigonometric and exponential forms of a complex number. Operations with complex numbers. De Moivre's formula. The n-th roots of a complex number. The Fundamental Theorem of Algebra and its consequences.

Sets, functions and metric spaces

Basic topics on sets and functions. Equipotent sets. Finite, countable and uncountable sets. Uncountability of R. The normed vector space R^n. Cauchy-Schwartz' inequality. Metric spaces and their topology: bounded, open, closed, compact and connected sets. Compactification of R.

Sequences

Properties of convergent sequences in a metric space. Cauchy sequences. Subsequences. Sequences of real numbers. Limits and their operations. Monotone sequences. The number e. Special limits.

Numerical series

Convergence and divergence of numerical series. Absolute convergence. Cauchy's criterion for convergence. Sufficient criteria for absolute convergence. Alternating series and the Leibnitz criterion.

Mappings between metric spaces

Limits of functions: metric and sequential definitions. Pointwise and global continuity. Inverse images of open sets Continuity, compactness, connectedness. Continuity, composition and invertibility. Uniform continuity. Real functions of one real variable. Limits for monotonic functions. Asymptotic behaviour. Discontinuities.

Differential calculus for real functions of one real variable

Differentiability: geometrical meaning and continuity. Differentiation rules. The derivatives of elementary functions. Higher derivatives. Differentiability, composition, invertibility. The theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. L'Hospital's theorems. Taylor's formula with the Peano and Lagrange forms for the remainder. Mac-Laurin's formula for elementary functions. Maxima and minima. Convexity in an interval. Inflection points.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8

Practicals: 40 hours

Lessons: 32 hours

Lessons: 32 hours

Professor(s)