#
Mathematical analysis 1

A.Y. 2020/2021

Learning objectives

The course aims to provide the student with an introduction the beginnings of a deeper understanding of Mathematical Analysis (essential for a student of physics) which will continue in the courses Mathematical Analysis 2 and 3. Fundamental concepts will be treated with rigor and precision with particular attention dedicated to: the real number system viewed as a complete ordered field with the cardinality of the continuum, metric spaces (including the real numbers) viewed as an abstract and general setting in order to formulate in a robust way the notions of limits of sequences in metric spaces, limits and continuity of functions between metric spaces (fundamental for the development of differential and integral calculus in Euclidean spaces), and, reducing to the important special case of the real number field, convergence of sequences and series and differential calculus for real functions of a real variable.

Expected learning outcomes

1. Knowledge and understanding of the concept of a complete ordered field with the cardinality of the continuum.

2. Knowledge and understanding of the concept of a metric space and elements of the topology of metric spaces (classification of points and sets)

3. Knowledge and understanding of the concept of the limit of a sequence in a metric space (uniqueness of the limit, necessary conditions for the convergence).

4. Knowledge and understanding of the concept of the limit of a sequence of real numbers and techniques for the calculation of limits (regularity of monotone sequences, comparison criteria, algebraic properties, Landau symbols e comparison for vanishing and diverging sequences).

5. Knowledge and understanding of the concept of convergence and divergence for numerical series (modes of convergence and relations between them).

6. Knowledge and understanding of techniques for establishing the character of a series (Cauchy's criterion, ratio test, root test, Leibniz's test, condensation test, for example).

7. Knowledge and understanding of the concepts of limit and continuity for functions between metric spaces.

8. Knowledge and understanding of the consequences of continuity (existence for equations involving real valued functions, existence of maxima and minima for real value functions on compact sets, the intermediate value property for real valued functions).

9. Knowledge and understanding of the concept of differentiability for real valued functions of a real variable (geometric and physical significance for unidimensional motions of the first and second derivatives).

10. Knowledge and understanding of techniques for the calculation of derivatives (algebraic properties, compositions and the chain rule, derivative of inverse functions).

11. Knowledge and understanding of the applications of differential calculus for real valued functions of a real variable (monotonicity, convexity, local maximums and minimums, qualitative study of functions).

12. Ability to correctly state the principal definitions and theorems.

13. Ability to perform abstract reasoning in concrete situations.

14. Ability to perform quickly and correctly calculations in the resolution of exercises in the written exams.

15. Ability to think synthetically and critically with regard to the concepts studied and the relations between them.

16. Ability to think critically concerning the value of the mathematics studied in the context of the degree program in physics.

17. Ability to communicate in the discussions of the material studied during the final oral examination.

18. Ability to continue in an autonomous way a deeper understanding of mathematical analysis in future courses.

19. Possibility to work in groups during the recitation sections, tutorial and in preparation for the written and oral examinations.

2. Knowledge and understanding of the concept of a metric space and elements of the topology of metric spaces (classification of points and sets)

3. Knowledge and understanding of the concept of the limit of a sequence in a metric space (uniqueness of the limit, necessary conditions for the convergence).

4. Knowledge and understanding of the concept of the limit of a sequence of real numbers and techniques for the calculation of limits (regularity of monotone sequences, comparison criteria, algebraic properties, Landau symbols e comparison for vanishing and diverging sequences).

5. Knowledge and understanding of the concept of convergence and divergence for numerical series (modes of convergence and relations between them).

6. Knowledge and understanding of techniques for establishing the character of a series (Cauchy's criterion, ratio test, root test, Leibniz's test, condensation test, for example).

7. Knowledge and understanding of the concepts of limit and continuity for functions between metric spaces.

8. Knowledge and understanding of the consequences of continuity (existence for equations involving real valued functions, existence of maxima and minima for real value functions on compact sets, the intermediate value property for real valued functions).

9. Knowledge and understanding of the concept of differentiability for real valued functions of a real variable (geometric and physical significance for unidimensional motions of the first and second derivatives).

10. Knowledge and understanding of techniques for the calculation of derivatives (algebraic properties, compositions and the chain rule, derivative of inverse functions).

11. Knowledge and understanding of the applications of differential calculus for real valued functions of a real variable (monotonicity, convexity, local maximums and minimums, qualitative study of functions).

12. Ability to correctly state the principal definitions and theorems.

13. Ability to perform abstract reasoning in concrete situations.

14. Ability to perform quickly and correctly calculations in the resolution of exercises in the written exams.

15. Ability to think synthetically and critically with regard to the concepts studied and the relations between them.

16. Ability to think critically concerning the value of the mathematics studied in the context of the degree program in physics.

17. Ability to communicate in the discussions of the material studied during the final oral examination.

18. Ability to continue in an autonomous way a deeper understanding of mathematical analysis in future courses.

19. Possibility to work in groups during the recitation sections, tutorial and in preparation for the written and oral examinations.

**Lesson period:** First semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

First semester

The lectures and problem sessions (esercitazioni) will be conducted remotely

in virtual classrooms (on the zoom platform) via live streaming, allowing for real-time interaction between the students and the instructor.

In addition, it is expected that there will be an enrichment activity conducted in person. Groups of students will meet with an instructor once every two weeks. More detailed information concerning this enrichment activity will be published online on the Ariel platform.

in virtual classrooms (on the zoom platform) via live streaming, allowing for real-time interaction between the students and the instructor.

In addition, it is expected that there will be an enrichment activity conducted in person. Groups of students will meet with an instructor once every two weeks. More detailed information concerning this enrichment activity will be published online on the Ariel platform.

**Course syllabus**

1. The real number system

Known sets of numbers: decimal representation of rational numbers. Real numbers: definition and fundamental properties. The least upper bound and greatest lower bound. Completeness of the real numbers. Euclidian spaces.

2. Elements of set theory and metric spaces

Functions and their elementary properties. Cardinality of sets: finite, countably infinite and uncountably infinite sets. The uncountability of the real numbers. Metric spaces: neighborhoods and classification of points. Types of sets in metric spaces (open, closed, bounded, compact) and their properties. Nested families of closed intervals and the theorem of Bolzano-Weierstrass.

3. Sequences

Convergence of sequences in metric spaces: definition and fundamental properties. Uniqueness of limits and boundedness of convergent sequences. The Cauchy criterion and the completeness of Euclidian spaces. Real valued sequences: operations, sign preservation under limits and comparison tests. Monotonicity and existence of limits. The number e of Napier and notable limit formulas. Subsequences: compactness and the limits of subsequences. Landau symbols for comparison of asymptotic behavior of sequences.

4. Numerical series

Definitions, examples and character of infinite series of real numbers. The Cauchy criterion and necessary conditions for convergence. Absolute convergence. Tests for convergence of series with non-negative terms: comparison, limit comparison, ratio and roots tests. The condensation test. Alternating series and the Leibniz criterion.

5. Limits and continuity of functions

Mappings between metric spaces. Metric and sequence definitions for the existence of limits and their equivalence. Limits of elementary functions and the existence of limits for real valued functions of a real variable by monotonicity. Asymptotes to the graph of a function. Pointwise and global continuity. Continuity and the inverse image of open sets. Composition of continuous functions. Continuity, compactness and the theorem of Weierstrass. Uniform continuity and the theorem of Heine-Cantor. Real valued functions: the theorems on zeros, intermediate values and Darboux's theorem. Monotonicity and continuity. Continuity of the inverse function.

6. Differential calculus for real valued functions of a real variable

The derivative: definition and geometric interpretation in terms of the tangent line. Continuity and differentiability. Derivative of elementary functions and differentiation rules for algebraic operations, compositions and inversion. Higher order derivatives. Local extrema: the theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. Calculus of indefinite form limits by the theorem of de l'Hôpital. Taylor's formula of second order with Peano and Lagrange remainders. Taylor expansions of elementary functions. Convexity, concavity and inflection points. Applications to local extrema of functions.

Known sets of numbers: decimal representation of rational numbers. Real numbers: definition and fundamental properties. The least upper bound and greatest lower bound. Completeness of the real numbers. Euclidian spaces.

2. Elements of set theory and metric spaces

Functions and their elementary properties. Cardinality of sets: finite, countably infinite and uncountably infinite sets. The uncountability of the real numbers. Metric spaces: neighborhoods and classification of points. Types of sets in metric spaces (open, closed, bounded, compact) and their properties. Nested families of closed intervals and the theorem of Bolzano-Weierstrass.

3. Sequences

Convergence of sequences in metric spaces: definition and fundamental properties. Uniqueness of limits and boundedness of convergent sequences. The Cauchy criterion and the completeness of Euclidian spaces. Real valued sequences: operations, sign preservation under limits and comparison tests. Monotonicity and existence of limits. The number e of Napier and notable limit formulas. Subsequences: compactness and the limits of subsequences. Landau symbols for comparison of asymptotic behavior of sequences.

4. Numerical series

Definitions, examples and character of infinite series of real numbers. The Cauchy criterion and necessary conditions for convergence. Absolute convergence. Tests for convergence of series with non-negative terms: comparison, limit comparison, ratio and roots tests. The condensation test. Alternating series and the Leibniz criterion.

5. Limits and continuity of functions

Mappings between metric spaces. Metric and sequence definitions for the existence of limits and their equivalence. Limits of elementary functions and the existence of limits for real valued functions of a real variable by monotonicity. Asymptotes to the graph of a function. Pointwise and global continuity. Continuity and the inverse image of open sets. Composition of continuous functions. Continuity, compactness and the theorem of Weierstrass. Uniform continuity and the theorem of Heine-Cantor. Real valued functions: the theorems on zeros, intermediate values and Darboux's theorem. Monotonicity and continuity. Continuity of the inverse function.

6. Differential calculus for real valued functions of a real variable

The derivative: definition and geometric interpretation in terms of the tangent line. Continuity and differentiability. Derivative of elementary functions and differentiation rules for algebraic operations, compositions and inversion. Higher order derivatives. Local extrema: the theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. Calculus of indefinite form limits by the theorem of de l'Hôpital. Taylor's formula of second order with Peano and Lagrange remainders. Taylor expansions of elementary functions. Convexity, concavity and inflection points. Applications to local extrema of functions.

**Prerequisites for admission**

Elementary algebra and geometry; knowledge and good use of trigonometry. Exponential and logarithmic functions and their applications to equations and inequalities.

**Teaching methods**

In order to facilitate the learning process, classroom lectures and problem sessions will be offered to the students, who are strongly encouraged to participate in these activities. In addition, weekly tutorial sessions will be offered in order to give the students the possibility to interact directly with the tutor in order to discuss the solution of weekly homework exercises which will be assigned by the instructors of the course.

**Teaching Resources**

P.M. Soardi. Analisi Matematica. Città Studi.

M. Amar, A.M. Bersani. Esercizi di Analisi matematica. Progetto Leonardo.

L. De Michele e G.L. Forti, Analisi Matematica Problemi ed Esercizi, Clup.

M. Amar, A.M. Bersani. Esercizi di Analisi matematica. Progetto Leonardo.

L. De Michele e G.L. Forti, Analisi Matematica Problemi ed Esercizi, Clup.

**Assessment methods and Criteria**

The exam consists of a written and an oral part.

The written exam usually lasts 120 minutes. It aims to verify the knowledge and understanding of the various concepts and techniques in concrete situations, through the solution of exercises, focused on: the real number field, metric spaces, sequences and series of real numbers, differential calculus for real valued functions of a real variable, convergence of sequences and series and differential calculus for real functions of a real variable. The complete written exam can be taken in the form of two partial exams (one mid-term exam plus a second partial exam to be offered in one of the first two exam dates of the winter session in January and February) which together will cover the topics of the complete exam.

The final oral exam, which can be taken upon successful completion of the written exam, consists in a discussion of the arguments developed in the course and/or treated in the written exam. The purpose of the oral exam is to evaluate the knowledge acquired on the theoretical aspects of the material developed and on the capacity to apply this knowledge in the solution of concrete problems, including situations not necessarily examined explicitly during the course. The final evaluation will be quantified with 30 as the maximum grade.

The written exam usually lasts 120 minutes. It aims to verify the knowledge and understanding of the various concepts and techniques in concrete situations, through the solution of exercises, focused on: the real number field, metric spaces, sequences and series of real numbers, differential calculus for real valued functions of a real variable, convergence of sequences and series and differential calculus for real functions of a real variable. The complete written exam can be taken in the form of two partial exams (one mid-term exam plus a second partial exam to be offered in one of the first two exam dates of the winter session in January and February) which together will cover the topics of the complete exam.

The final oral exam, which can be taken upon successful completion of the written exam, consists in a discussion of the arguments developed in the course and/or treated in the written exam. The purpose of the oral exam is to evaluate the knowledge acquired on the theoretical aspects of the material developed and on the capacity to apply this knowledge in the solution of concrete problems, including situations not necessarily examined explicitly during the course. The final evaluation will be quantified with 30 as the maximum grade.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8

Practicals: 48 hours

Lessons: 32 hours

Lessons: 32 hours

Professors:
Calanchi Marta, Terraneo Elide

### CORSO B

Responsible

Lesson period

First semester

in virtual classrooms (on the zoom platform) via live streaming, allowing for real-time interaction between the students and the instructor.

In addition, it is expected that there will be an enrichment activity conducted in person. Groups of students will meet with an instructor once every two weeks. More detailed information concerning this enrichment activity will be published online on the Ariel platform.

**Course syllabus**

1. The real number system

Known sets of numbers: decimal representation of rational numbers. Real numbers: definition and fundamental properties. The least upper bound and greatest lower bound. Completeness of the real numbers. Euclidian spaces.

2. Elements of set theory and metric spaces

Functions and their elementary properties. Cardinality of sets: finite, countably infinite and uncountably infinite sets. The uncountability of the real numbers. Metric spaces: neighborhoods and classification of points. Types of sets in metric spaces (open, closed, bounded, compact) and their properties. Nested families of closed intervals and the theorem of Bolzano-Weierstrass.

3. Sequences

Convergence of sequences in metric spaces: definition and fundamental properties. Uniqueness of limits and boundedness of convergent sequences. The Cauchy criterion and the completeness of Euclidian spaces. Real valued sequences: operations, sign preservation under limits and comparison tests. Monotonicity and existence of limits. The number e of Napier and notable limit formulas. Subsequences: compactness and the limits of subsequences. Landau symbols for comparison of asymptotic behavior of sequences.

4. Numerical series

Definitions, examples and character of infinite series of real numbers. The Cauchy criterion and necessary conditions for convergence. Absolute convergence. Tests for convergence of series with non-negative terms: comparison, limit comparison, ratio and roots tests. The condensation test. Alternating series and the Leibniz criterion.

5. Limits and continuity of functions

Mappings between metric spaces. Metric and sequence definitions for the existence of limits and their equivalence. Limits of elementary functions and the existence of limits for real valued functions of a real variable by monotonicity. Asymptotes to the graph of a function. Pointwise and global continuity. Continuity and the inverse image of open sets. Composition of continuous functions. Continuity, compactness and the theorem of Weierstrass. Uniform continuity and the theorem of Heine-Cantor. Real valued functions: the theorems on zeros, intermediate values and Darboux's theorem. Monotonicity and continuity. Continuity of the inverse function.

6. Differential calculus for real valued functions of a real variable

The derivative: definition and geometric interpretation in terms of the tangent line. Continuity and differentiability. Derivative of elementary functions and differentiation rules for algebraic operations, compositions and inversion. Higher order derivatives. Local extrema: the theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. Calculus of indefinite form limits by the theorem of de l'Hôpital. Taylor's formula of second order with Peano and Lagrange remainders. Taylor expansions of elementary functions. Convexity, concavity and inflection points. Applications to local extrema of functions.

Known sets of numbers: decimal representation of rational numbers. Real numbers: definition and fundamental properties. The least upper bound and greatest lower bound. Completeness of the real numbers. Euclidian spaces.

2. Elements of set theory and metric spaces

Functions and their elementary properties. Cardinality of sets: finite, countably infinite and uncountably infinite sets. The uncountability of the real numbers. Metric spaces: neighborhoods and classification of points. Types of sets in metric spaces (open, closed, bounded, compact) and their properties. Nested families of closed intervals and the theorem of Bolzano-Weierstrass.

3. Sequences

Convergence of sequences in metric spaces: definition and fundamental properties. Uniqueness of limits and boundedness of convergent sequences. The Cauchy criterion and the completeness of Euclidian spaces. Real valued sequences: operations, sign preservation under limits and comparison tests. Monotonicity and existence of limits. The number e of Napier and notable limit formulas. Subsequences: compactness and the limits of subsequences. Landau symbols for comparison of asymptotic behavior of sequences.

4. Numerical series

Definitions, examples and character of infinite series of real numbers. The Cauchy criterion and necessary conditions for convergence. Absolute convergence. Tests for convergence of series with non-negative terms: comparison, limit comparison, ratio and roots tests. The condensation test. Alternating series and the Leibniz criterion.

5. Limits and continuity of functions

Mappings between metric spaces. Metric and sequence definitions for the existence of limits and their equivalence. Limits of elementary functions and the existence of limits for real valued functions of a real variable by monotonicity. Asymptotes to the graph of a function. Pointwise and global continuity. Continuity and the inverse image of open sets. Composition of continuous functions. Continuity, compactness and the theorem of Weierstrass. Uniform continuity and the theorem of Heine-Cantor. Real valued functions: the theorems on zeros, intermediate values and Darboux's theorem. Monotonicity and continuity. Continuity of the inverse function.

6. Differential calculus for real valued functions of a real variable

The derivative: definition and geometric interpretation in terms of the tangent line. Continuity and differentiability. Derivative of elementary functions and differentiation rules for algebraic operations, compositions and inversion. Higher order derivatives. Local extrema: the theorems of Fermat, Rolle, Cauchy, Lagrange and their consequences. Calculus of indefinite form limits by the theorem of de l'Hôpital. Taylor's formula of second order with Peano and Lagrange remainders. Taylor expansions of elementary functions. Convexity, concavity and inflection points. Applications to local extrema of functions.

**Prerequisites for admission**

Elementary algebra and geometry; knowledge and good use of trigonometry. Exponential and logarithmic functions and their applications to equations and inequalities.

**Teaching methods**

In order to facilitate the learning process, classroom lectures and problem sessions will be offered to the students, who are strongly encouraged to participate in these activities. In addition, weekly tutorial sessions will be offered in order to give the students the possibility to interact directly with the tutor in order to discuss the solution of weekly homework exercises which will be assigned by the instructors of the course.

**Teaching Resources**

P.M. Soardi. Analisi Matematica. Città Studi.

M. Amar, A.M. Bersani. Esercizi di Analisi matematica. Progetto Leonardo.

L. De Michele e G.L. Forti, Analisi Matematica Problemi ed Esercizi, Clup.

M. Amar, A.M. Bersani. Esercizi di Analisi matematica. Progetto Leonardo.

L. De Michele e G.L. Forti, Analisi Matematica Problemi ed Esercizi, Clup.

**Assessment methods and Criteria**

The exam consists of a written and an oral part.

The written exam usually lasts 120 minutes. It aims to verify the knowledge and understanding of the various concepts and techniques in concrete situations, through the solution of exercises, focused on: the real number field, metric spaces, sequences and series of real numbers, differential calculus for real valued functions of a real variable, convergence of sequences and series and differential calculus for real functions of a real variable. The complete written exam can be taken in the form of two partial exams (one mid-term exam plus a second partial exam to be offered in one of the first two exam dates of the winter session in January and February) which together will cover the topics of the complete exam.

The final oral exam, which can be taken upon successful completion of the written exam, consists in a discussion of the arguments developed in the course and/or treated in the written exam. The purpose of the oral exam is to evaluate the knowledge acquired on the theoretical aspects of the material developed and on the capacity to apply this knowledge in the solution of concrete problems, including situations not necessarily examined explicitly during the course. The final evaluation will be quantified with 30 as the maximum grade.

The written exam usually lasts 120 minutes. It aims to verify the knowledge and understanding of the various concepts and techniques in concrete situations, through the solution of exercises, focused on: the real number field, metric spaces, sequences and series of real numbers, differential calculus for real valued functions of a real variable, convergence of sequences and series and differential calculus for real functions of a real variable. The complete written exam can be taken in the form of two partial exams (one mid-term exam plus a second partial exam to be offered in one of the first two exam dates of the winter session in January and February) which together will cover the topics of the complete exam.

The final oral exam, which can be taken upon successful completion of the written exam, consists in a discussion of the arguments developed in the course and/or treated in the written exam. The purpose of the oral exam is to evaluate the knowledge acquired on the theoretical aspects of the material developed and on the capacity to apply this knowledge in the solution of concrete problems, including situations not necessarily examined explicitly during the course. The final evaluation will be quantified with 30 as the maximum grade.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 8

Practicals: 48 hours

Lessons: 32 hours

Lessons: 32 hours

Professors:
Cozzi Matteo, Payne Kevin Ray

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