#
Continuum mathematics

A.Y. 2018/2019

Learning objectives

A) Fornire le conoscenze di base relative ai numeri reali e complessi, e alcuni rudimenti di algebra lineare.

B) Introdurre alcune funzioni elementari e i concetti di base del calcolo differenziale e integrale, soprattutto per le funzioni reali (o complesse) di una variabile reale.

B) Introdurre alcune funzioni elementari e i concetti di base del calcolo differenziale e integrale, soprattutto per le funzioni reali (o complesse) di una variabile reale.

Expected learning outcomes

Undefined

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

**ATTENDING STUDENTS**

**Course syllabus**

1. Basic concepts of set theory. Relations on sets. Mappings between sets. Basics of

enumerative combinatorics.

2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their

algebraic structures and order relations. Completeness of 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. The notions of limit and

continuity for such functions. Some elementary functions: polynomials, exponential,

logarithm, trigonometric functions.

5. Real sequences and their limits. Real series.

6. The derivative of a real function of one real variable . A discussion on the geometrical

meaning and on the applications of the derivative. Derivatives of elementary functions.

Basic facts about derivable functions.

7. Higher order derivatives. Taylor's formula. Use of Taylor's formula in the computation of

limits. Taylor's series.

8. How to use the derivatives to determine the maximum and minimum points of a real

function of one real variable, as well as the intervals where the function is increasing,

decreasing, convex or concave.

9. The theory of Riemann's integral for real functions of one real variable. Geometrical

meaning of Riemann's integral. The ''fundamental theorem of calculus''. Basic integration

rules.

10. 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.

enumerative combinatorics.

2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their

algebraic structures and order relations. Completeness of 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. The notions of limit and

continuity for such functions. Some elementary functions: polynomials, exponential,

logarithm, trigonometric functions.

5. Real sequences and their limits. Real series.

6. The derivative of a real function of one real variable . A discussion on the geometrical

meaning and on the applications of the derivative. Derivatives of elementary functions.

Basic facts about derivable functions.

7. Higher order derivatives. Taylor's formula. Use of Taylor's formula in the computation of

limits. Taylor's series.

8. How to use the derivatives to determine the maximum and minimum points of a real

function of one real variable, as well as the intervals where the function is increasing,

decreasing, convex or concave.

9. The theory of Riemann's integral for real functions of one real variable. Geometrical

meaning of Riemann's integral. The ''fundamental theorem of calculus''. Basic integration

rules.

10. 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.

**NON-ATTENDING STUDENTS**

**Course syllabus**

1. Basic concepts of set theory. Relations on sets. Mappings between sets. Basics of

enumerative combinatorics.

2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their

algebraic structures and order relations. Completeness of 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. The notions of limit and

continuity for such functions. Some elementary functions: polynomials, exponential,

logarithm, trigonometric functions.

5. Real sequences and their limits. Real series.

6. The derivative of a real function of one real variable . A discussion on the geometrical

meaning and on the applications of the derivative. Derivatives of elementary functions.

Basic facts about derivable functions.

7. Higher order derivatives. Taylor's formula. Use of Taylor's formula in the computation of

limits. Taylor's series.

8. How to use the derivatives to determine the maximum and minimum points of a real

function of one real variable, as well as the intervals where the function is increasing,

decreasing, convex or concave.

9. The theory of Riemann's integral for real functions of one real variable. Geometrical

meaning of Riemann's integral. The ''fundamental theorem of calculus''. Basic integration

rules.

10. 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.

enumerative combinatorics.

2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their

algebraic structures and order relations. Completeness of 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. The notions of limit and

continuity for such functions. Some elementary functions: polynomials, exponential,

logarithm, trigonometric functions.

5. Real sequences and their limits. Real series.

6. The derivative of a real function of one real variable . A discussion on the geometrical

meaning and on the applications of the derivative. Derivatives of elementary functions.

Basic facts about derivable functions.

7. Higher order derivatives. Taylor's formula. Use of Taylor's formula in the computation of

limits. Taylor's series.

8. How to use the derivatives to determine the maximum and minimum points of a real

function of one real variable, as well as the intervals where the function is increasing,

decreasing, convex or concave.

9. The theory of Riemann's integral for real functions of one real variable. Geometrical

meaning of Riemann's integral. The ''fundamental theorem of calculus''. Basic integration

rules.

10. 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.

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