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Continuum mathematics

A.Y. 2019/2020

Learning objectives

The aim of the course is to provide a basic knowledge of the following subjects: the general methods of mathematical thinking; elementary set theory; the main number systems and their algebraic and order structures; linear algebra; some elementary functions of one real (or complex) variable; the notion of limit, differential calculus and integral calculus, mainly for real (or complex) functions of one real variable; the use of elementary functions and infinitesimal calculus in some real world applications.

Expected learning outcomes

The student should master the general methods of mathematical reasoning; he/she should attain a deep understanding of the basic theoretical notions provided by the course, and develop the capability to illustrate them in a rational way. In addition, the student should acquire the skill to solve computational problems in the same areas, applying autonomously the solution techniques provided by the course.

**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

### Single session

Responsible

Lesson period

First semester

**Course syllabus**

1. Elementary set theory. Basic notions on maps

between sets. Relations on sets. Equivalence

and order relations. Basics of enumerative combinatorics.

2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their algebraic and order structures. Countability of Q, uncountability of R. Completeness of R. Supremum and infimum of a subset of R. Elementary topological notions in R. The spaces R^n (n=1,2,3, ).

3. A sketch of the abstract notion of vector space. Linear mappings between vector spaces and matrices. Linear equations.

4. Some generalities on real functions from a subset of R to R. Some elementary functions: power functions, exponentials, logarithms, trigonometric functions. Using

trigonometric functions in acoustics. An invitation to Fourier series.

5. The notions of limit and continuity for functions from a subset of R to R. Some limits concerning elementary functions. Limits and algebraic operations on functions:

indeterminate forms. Limits and composition of functions. Notable special limits. Theorems of Darboux and Weierstrass on continuous functions.

6. Real sequences and their limits. Real series: some examples and convergence criteria.

7. The notion of derivative for a function from a subset of R to R, and its geometrical meaning. An application: the velocity of a particle. Derivatives and algebraic operations on functions. Derivatives of an inverse function and of the composition of two functions. The theorems of Rolle, Cauchy and Lagrange.

8. Higher order derivatives. An application of the second derivative: acceleration of a particle. Taylor's formula with reminder in the Peano or in the Lagrange form. Use of Taylor's formula in the calculation of limits, and in the numerical computation of functions. Taylor's series.

9. Using derivatives to determine the maximum and minimum points of a function from a subset of R to R, as well as the intervals where the function is increasing, decreasing, convex or concave. Analyzing other aspects of the graph of such a function; asymptotes.

10. The notion of primitive function, or indefinite integral, for a real function on an interval. Some elementary indefinite integrals. Integration by parts and by substitution.

11. The notion of Riemann's definite integral for real functions on an interval, and its geometrical meaning. The fundamental theorem of calculus. Integration by parts and by substitution in the definite case. An introduction to improper integrals. Using integrals to estimate finite sums and series.

12. The field C of complex numbers. Modulus, argument and trigonometric representation of a complex number. A sketch of the notions of limit, derivative and integral for complex valued functions. Complex sequences and series. The exponential function in the complex

field; Euler's formula. Roots of complex numbers. The fundamental theorem of algebra.

between sets. Relations on sets. Equivalence

and order relations. Basics of enumerative combinatorics.

2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their algebraic and order structures. Countability of Q, uncountability of R. Completeness of R. Supremum and infimum of a subset of R. Elementary topological notions in R. The spaces R^n (n=1,2,3, ).

3. A sketch of the abstract notion of vector space. Linear mappings between vector spaces and matrices. Linear equations.

4. Some generalities on real functions from a subset of R to R. Some elementary functions: power functions, exponentials, logarithms, trigonometric functions. Using

trigonometric functions in acoustics. An invitation to Fourier series.

5. The notions of limit and continuity for functions from a subset of R to R. Some limits concerning elementary functions. Limits and algebraic operations on functions:

indeterminate forms. Limits and composition of functions. Notable special limits. Theorems of Darboux and Weierstrass on continuous functions.

6. Real sequences and their limits. Real series: some examples and convergence criteria.

7. The notion of derivative for a function from a subset of R to R, and its geometrical meaning. An application: the velocity of a particle. Derivatives and algebraic operations on functions. Derivatives of an inverse function and of the composition of two functions. The theorems of Rolle, Cauchy and Lagrange.

8. Higher order derivatives. An application of the second derivative: acceleration of a particle. Taylor's formula with reminder in the Peano or in the Lagrange form. Use of Taylor's formula in the calculation of limits, and in the numerical computation of functions. Taylor's series.

9. Using derivatives to determine the maximum and minimum points of a function from a subset of R to R, as well as the intervals where the function is increasing, decreasing, convex or concave. Analyzing other aspects of the graph of such a function; asymptotes.

10. The notion of primitive function, or indefinite integral, for a real function on an interval. Some elementary indefinite integrals. Integration by parts and by substitution.

11. The notion of Riemann's definite integral for real functions on an interval, and its geometrical meaning. The fundamental theorem of calculus. Integration by parts and by substitution in the definite case. An introduction to improper integrals. Using integrals to estimate finite sums and series.

12. The field C of complex numbers. Modulus, argument and trigonometric representation of a complex number. A sketch of the notions of limit, derivative and integral for complex valued functions. Complex sequences and series. The exponential function in the complex

field; Euler's formula. Roots of complex numbers. The fundamental theorem of algebra.

**Prerequisites for admission**

The course takes place in the initial semester of the first year; consequently, there are no prerequisites except for those required for admission to the degree programme in

Music Information Science.

Music Information Science.

**Teaching methods**

The teaching is based on classroom lectures. During the lectures concerning theoretical aspects the teacher shows and accurately comments the slides of his notes, which are also available on the web page of the course.

**Teaching Resources**

The contents of all lectures about theoretical aspects

are exhaustively described by the notes on the course

written by the teacher, available on the web page of the course at the address

http://www.mat.unimi.it/users/pizzocchero/ .

The notes mentioned above also exemplify the resolution of computational problems on the same subjects.

Further material on computational aspects is available

on the sites ''Minimat'' e ''Matematica assistita'' , which can be accessed at the address

https://ariel.unimi.it/

Just for completeness, some classical textbooks on the topics of the course are listed hereafter.

● T. Apostol, "Calculus Volume I. One-variable Calculus, with an Introduction to Linear Algebra", Wiley India;

● A. Avantaggiati, ''Istituzioni di matematica'', Ed. Ambrosiana;

● G.C. Barozzi, ''Primo corso di analisi matematica'', Ed. Zanichelli;

● G.C. Barozzi, C. Corradi, ''Matematica generale per le scienze economiche'', Ed. Il Mulino;

● A. Guerraggio, ''Matematica generale'', Ed. Bollati Boringhieri.

are exhaustively described by the notes on the course

written by the teacher, available on the web page of the course at the address

http://www.mat.unimi.it/users/pizzocchero/ .

The notes mentioned above also exemplify the resolution of computational problems on the same subjects.

Further material on computational aspects is available

on the sites ''Minimat'' e ''Matematica assistita'' , which can be accessed at the address

https://ariel.unimi.it/

Just for completeness, some classical textbooks on the topics of the course are listed hereafter.

● T. Apostol, "Calculus Volume I. One-variable Calculus, with an Introduction to Linear Algebra", Wiley India;

● A. Avantaggiati, ''Istituzioni di matematica'', Ed. Ambrosiana;

● G.C. Barozzi, ''Primo corso di analisi matematica'', Ed. Zanichelli;

● G.C. Barozzi, C. Corradi, ''Matematica generale per le scienze economiche'', Ed. Il Mulino;

● A. Guerraggio, ''Matematica generale'', Ed. Bollati Boringhieri.

**Assessment methods and Criteria**

The examination consists of a written and an oral exam.

The written exam can be taken in the complete version,

or in two partial versions. The first one of these two

partial written exams is taken at a fixed date while the course is being delivered; the second partial written exam can be taken during one of first two examination sessions after the end of the course.

During a partial or complete written exam the student must solve some computational problems, formulated as open-ended questions.

The time duration of each partial written exam is two

hours; the duration of a complete written exam

is three hours. Each partial or complete written exam

receives a rating which can be, e.g., "not sufficient",

"almost sufficient", "sufficient", "good", "excellent"; this rating is communicated to the student immediately after the correction (typically, few days after the date of the exam).

The oral exam can be taken only if the complete written component or both partial written components have been qualified, at least, as "almost sufficient".

During the oral exam the student is demanded to show that he/she masters the main theoretical concepts illustrated in the course, including the proofs of the main theorems, and that he/she is able to expose them rationally. Again during the oral exam, at the discretion

of the examiners, the student can be invited to

discuss his/her written exam and, possibly, to solve a computational problem similar to one that he/she has not managed adequately during the written exam.

The complete final exam is passed if both the written and oral parts are passed.

Final marks are given using the numerical range 0-30, and taking into account both the written and the oral parts. These final marks are communicated immediately after the oral exam.

For further information about examinations, see the web page of the course; this reports the texts of all problems

assigned for written exams during the last years, and

also makes available a guide to the preparation of the oral exam.

The written exam can be taken in the complete version,

or in two partial versions. The first one of these two

partial written exams is taken at a fixed date while the course is being delivered; the second partial written exam can be taken during one of first two examination sessions after the end of the course.

During a partial or complete written exam the student must solve some computational problems, formulated as open-ended questions.

The time duration of each partial written exam is two

hours; the duration of a complete written exam

is three hours. Each partial or complete written exam

receives a rating which can be, e.g., "not sufficient",

"almost sufficient", "sufficient", "good", "excellent"; this rating is communicated to the student immediately after the correction (typically, few days after the date of the exam).

The oral exam can be taken only if the complete written component or both partial written components have been qualified, at least, as "almost sufficient".

During the oral exam the student is demanded to show that he/she masters the main theoretical concepts illustrated in the course, including the proofs of the main theorems, and that he/she is able to expose them rationally. Again during the oral exam, at the discretion

of the examiners, the student can be invited to

discuss his/her written exam and, possibly, to solve a computational problem similar to one that he/she has not managed adequately during the written exam.

The complete final exam is passed if both the written and oral parts are passed.

Final marks are given using the numerical range 0-30, and taking into account both the written and the oral parts. These final marks are communicated immediately after the oral exam.

For further information about examinations, see the web page of the course; this reports the texts of all problems

assigned for written exams during the last years, and

also makes available a guide to the preparation of the oral exam.

MAT/01 - MATHEMATICAL LOGIC - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

MAT/02 - ALGEBRA - University credits: 0

MAT/03 - GEOMETRY - University credits: 0

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0

MAT/06 - PROBABILITY AND STATISTICS - University credits: 0

MAT/07 - MATHEMATICAL PHYSICS - University credits: 0

MAT/08 - NUMERICAL ANALYSIS - University credits: 0

MAT/09 - OPERATIONS RESEARCH - University credits: 0

Practicals: 48 hours

Lessons: 64 hours

Lessons: 64 hours

Professor(s)

Reception:

Wednesday, 13.30-17.30

Room 1005, Department of Mathematics, Via Saldini 50, 20133, Milan