# MATHEMATICAL ANALYSIS

Academic year and teacher
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Versione italiana
2022/2023
Teacher
LORENZO BRASCO
Credits
12
Didactic period
Annuale
SSD
MAT/05

#### Training objectives

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Modulo: 64317 - ANALISI MATEMATICA A
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The aim of this course is to present the basic ideas and techniques of differential and integral calculus in one and several variables. We will also try, in spite of the few hours at our disposal, to give to the students something more than a working knowledge of mathematical techniques. Some problems in applied mathmatics will also be discussed and solved. At the end of the course the students must be able to solve problems of maximum and minimum (constrained and unconstrained), compute volumes and areas of surfaces, barycenters of solids and surfaces, compute the flux of a vector fields through a surface (closed or with boundary)

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Modulo: 64318 - ANALISI MATEMATICA B
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The aim of this course is to present the basic ideas and techniques of differential and integral calculus in one and several variables. We will also try, in spite of the few hours at our disposal, to give to the students something more than a working knowledge of mathematical techniques. Some problems in applied mathmatics will also be discussed and solved. At the end of the course the students must be able to solve problems of maximum and minimum (constrained and unconstrained), compute volumes and areas of surfaces, barycenters of solids and surfaces, compute the flux of a vector fields through a surface (closed or with boundary)

#### Prerequisites

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Modulo: 64317 - ANALISI MATEMATICA A
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The prerequisites are the basic elements of geometry, algebra and trigonometry which are usually taught in the high schools.

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Modulo: 64318 - ANALISI MATEMATICA B
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The prerequisites are the basic elements of geometry, algebra and trigonometry which are usually taught in the high schools.

#### Course programme

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Modulo: 64317 - ANALISI MATEMATICA A
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- Sets theory
- Number systems (naturals, integers, rationals and reals)
- Mathematical induction
- Sup and inf
- Functions (injectivity, surjectivity, invertibility)
- Elementary real functions (integer part, fractional part, absolute value, powers, exponentials, logarithms, trigonometric and hyperbolic functions)
- Numerical sequences
- Cesaro's Theorems for sequences
- Convergence tests for sequences
- Napier's constant
- Numerical series (geometric series and Riemann series)
- Convergence tests for numerical series with fixed sign (direct comparison, limit comparison, ratio, root, Cauchy condensation)
- Convergence tests for numerical series without fixed sign (absolute convergence, Leibniz' test)
- Limits of functions
- Theorems on limits
- Some important limits and asymptotic equivalences: sinx/x, (e^x-1)/x, (1+x)^{1/x}, log(1+x)/x, (1-cos x)/x^2, tanx/x
- Continuity
- Theorems on continuous functions: sign, Bolzano's Theorem on zeros of continuous functions, intermediate value theorem, Weierstrass' Theorem
- Differential calculus: derivatives and fundamental theorems (Fermat, Rolle, Cauchy, Lagrange and their corollaries)
- derivatives of higher order
- Taylor's formula (with Peano's remainder)
- Integral calculus: construction of Riemann's integral
- mean value Theorem
- integrability of continuous and monotone functions
- the fundamental Theorem
- Integration techniques (by parts and by substitution)
- Integral of a rational function
- Special change of variable: parametric formulas for trigonometric functions and hyperbolic functions (hyperbolic cosinus, sinus and tangent, their inverses)

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Modulo: 64318 - ANALISI MATEMATICA B
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- Curves in the space
- Smooth curves and rectifiable curves
- Reparametrizations
- Tangent versor
- Length of a curve
- Reparametrization by arc length
- Curves in the plane: cartesian form, polar form, normal versor, curvature
- Topology of R^N
- Graphs and level sets
- Limits and continuity for functions of several variables
- Computations of limits for functions of two variables
- The 2 Curves' Criterion for the limits
- Differential calculus for functions of several variables
- Partial derivatives
- Tangent plane and differentiability
- C^1 functions are differentiable
- Directional derivatives
- Schwarz's Theorem
- Second order Taylor formula for functions of two variables
- Hessian matrix
- Free optimization
- Study of critical points trough the Hessian matrix
- Constrained optimization: Lagrange's multipliers
- Regular surfaces
- Multiple integrals with applications to Mechanics
- Line integrals
- Barycenter and moment of inertia of a curve
- Integrals in R^2 and R^3
- Change of variables in the multiple integrals
- Barycenter and moment of inertia for open subsets of R^2 and R^3
- Surface integrals
- Area, barycenter and moment of inertia of a surface
- Vector fields
- Conservative vector fields and potentials
- Irrotational vector fields
- Necessary and sufficient conditions for a vector fields to be conservative
- Flux of a vector field
- The Divergence Theorem
- Solenoidal vector fields

#### Didactic methods

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Modulo: 64317 - ANALISI MATEMATICA A
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Lectures and exercices on all the subjects of the course.

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Modulo: 64318 - ANALISI MATEMATICA B
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Lectures and exercices on all the subjects of the course.

#### Learning assessment procedures

The examination consist of a written exam and an oral exam.

The written exam is divided in 2 parts, precisely:

- a first part with 10 questions, where the student is only asked to give the answer, without any justification. In case of a correct answer, the student gets 2 points. On the contrary, he does not get any point, in case of incomplete, incorrect or missing answer.
In this part, the questions concern simple maximum/minimum problems, computation of some integrals, computation of some limits (also with the aid of Taylor formula), study of a series, find and classify the critical points of a function, computation of a simple double integral, find the potential of a conservative field, recognize whether a vector field is conservative or not, computation of the work of vector field through a path.

- a second part, with 2 exercises: here the student is asked to give all the details for the solutions of each exercises. Both exercises worth 7 or 8 points. In this part, typically the exercises concern the study of a function of one variable, the computation of a multiple integral, the study of a regular surface, the computation of the flux of a vector field through a surface (closed or with boundary).

ATTENTION: the second part of the written exam will be checked only in case the student will be able to answer correctly at least 5 out of 10 of the questions in the first part. If this is not the case, the exam is concluded.

Whenever the final evaluation is greater than or equal to 15/30, the student has to pass an oral exam, with 2 theoretical questions on parts of the program. In particular, the 2 questions are taken from the following list:

1) definition of limit (for sequences and functions, of one and several real variables) and definition of continuous function (for functions of one and several real variables)
2) supremum and infimum
3) convergence criteria for sequences and series
4) geometric series and generalized harmonic series (definitions and convergence)
5) theorems of zeros and intermediate values
6) Weierstrass theorem (in one and several real variables)
7) Fermat, Rolle, Lagrange, Cauchy and de L'Hopital Theorems
8) Taylor formula
9) fundamental theorem of calculus
10) integrability criteria for functions of one variable
11) remarkable limits and asymptotics
12 length of a curve and rectifiability of regular curves
13) tangent versor, normal versor and curvature in the plane
14) differentiability for functions of several variables and sufficient conditions for differentiability
15) Hessian matrix and the Schwarz Theorem
16) gradient and directional derivatives
17) level lines of a function and their regularity (implicit function theorem)
18) critical points of a function of several variables and study of their nature by means of the hessian matrix
19) Lagrange multipliers theorem
20) double and triple integrals
21) line and surface integrals
22) conservative vector fields: definition and necessary/sufficient for a vector field to be conservative
23) divergence theorem

At the end of the oral exam, the student will get his final mark: if this is larger than or equal to 18/30, the exam is positively concluded. In case the final mark is less than 18/30, the student will have to pass again the complete exam (i.e. written and oral).

#### Reference texts

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Modulo: 64317 - ANALISI MATEMATICA A
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The book adopted is :

- M. Bramanti, C. D. Pagani, S. Salsa: MATEMATICA Calcolo Infinitesimale ed Algebra Lineare, Ed. Zanichelli (second edition)

On the web page of the course some lecture notes part A are available, together with the slides of all the lessons from the previous year.

For the part A of the course, we suggest the following book of exercises:

- P. Marcellini, C. Sbordone, Esercitazioni di Matematica (parte prima & parte seconda), Ed. Liguori

For the part B of the course, we suggest the book of exercises:

- P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica Due (I & II), Ed. Zanichelli

One could also consult:

- Lectures notes by Prof. Codecà (that can be found in the Copy Center of the Faculty)

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Modulo: 64318 - ANALISI MATEMATICA B
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The book adopted is :

- M. Bramanti, C. D. Pagani, S. Salsa: MATEMATICA Calcolo Infinitesimale ed Algebra Lineare, Ed. Zanichelli (second edition)

On the web page of the course some lecture notes part A are available, together with the slides of all the lessons from the previous year.

For the part A of the course, we suggest the following book of exercises:

- P. Marcellini, C. Sbordone, Esercitazioni di Matematica (parte prima & parte seconda), Ed. Liguori

For the part B of the course, we suggest the book of exercises:

- P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica Due (I & II), Ed. Zanichelli

One could also consult:

- Lectures notes by Prof. Codecà (that can be found in the Copy Center of the Faculty)