Mathematical analysis 1

A.Y. 2016/2017
8
Max ECTS
72
Overall hours
SSD
MAT/05
Language
Italian
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.
Expected learning outcomes
Undefined
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.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8
Practicals: 40 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.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8
Practicals: 40 hours
Lessons: 32 hours