# MATHEMATICAL ANALYSIS I.B

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Versione italiana
2022/2023
Teacher
ALESSIA ASCANELLI
Credits
6
Didactic period
Secondo Semestre
SSD
MAT/05

#### Training objectives

Aim of the course is to provide the basic tools of mathematical analysis, especially for what concerns differential calculus for multi-variable functions, ordinary differental equations, integral calculus for functions of one or more real variables and their applications to the resolution of problems based on mathematical models.
At the end of the unit, students will know the theoretical contents and the methods proper of mathematical analysis calculus.
They should be able to properly apply the concepts learnt to the resolution of different kinds of problems and to identify the most appropriate approach to do so.
Students will have to master the mathematical language and the logical-deductive approach, demonstrating the ability to illustrate their problem solving strategies in a logical, effective, pertinent and synthetic way.

#### Prerequisites

Full comprehension of all contents of the Analisi Matematica 1.A course is required.

#### Course programme

Chapter 1 RIEMANN'S INTEGRAL
Definition of Riemann's integral, lower and upper sums, integrable functions, geometric meaning of the integral.
Integrability criteria. Integrability of monotone functions and continuous functions. Properties of Riemann's integral: monotonicity, additivity, linearity. Integral mean value theorem. Integral functions. Primitives and indefinite integrals. The fundamental theorem of calculus.
Integration techniques: by direct recognition, by parts, by direct and reverse substitution, integration of Rational functions, some useful substitutions in the calculation of primitives.
Improper integrals. Convergence criteria for improper integrals .
Asymptotic comparison.

Chapter 2 DIFFERENTIAL CALCULUS FOR FUNCTIONS OF MULTIPLE VARIABLES
The space R^n. Real valued functions of multiple variables: domain, level sets, calculation of limits, continuity. Derivative in a direction. Partial Derivatives. Gradient. First order approximation and differentiation. Tangent plane to the graph of a function of multiple variables. Total differential theorem. Second derivatives and hessian matrix. Differential calculus for vector-valued functions. Derivatives, Jacobian matrix, divergence, rotor. Derivatives of composite functions of multiple variables.

Chapter 3 MULTIPLE INTEGRALS
Riemann's integral in R^n on Peano-Jordan measurable sets.
Double integrals on simple sets in the plane. Triple integrals by lines, by layers. Change of variable in multiple integrals. Polar, cylindrical, spherical coordinates. Solids of revolution and Guldino theorem for volumes. Calculation of volumes, center of gravity, moments of solids.
Multivariable improper integrals. Integration of the gaussian.

Chapter 4 COMPLEX NUMBERS
Algebra and geometry of complex numbers, complex exponential and logarithm, roots in the complex field, factorization of polynomials in the complex field, limits and derivatives of complex valued functions.

Chapter 5 ORDINARY DIFFERENTIAL EQUATIONS (ODEs)
What is an ordinary differential equation and what is a solution. The Cauchy problem. Lipschitz functions, existence and uniqueness of a local solution. ODEs with separable variables. First order homogeneous and non-homogeneous linear ODEs.
Second order homogeneous and non-homogeneous linear ODEs with constant coefficients.

#### Didactic methods

The course provides 60 hours of classroom lectures with presentation of theoretical aspects, applications and several exercises for each one of the topics of the course. Lectures are given on the blackboard/by Ipad.

At the end of each chapter the teacher gives to the students some exercises for home, to test the skills acquired; the same exercises will corrected during the lectures/sent to the students by Classroom about one week later, so that the students can have a feedback on the correctness or not of the exercises done.

#### Learning assessment procedures

The exam consists in:
- a written exam, aimed to assess the student's ability to solve problems and exercises;
- an oral exam, aimed at evaluating the theoretical knowledge and exhibition capacity.

During the written exam it is not permitted to consult texts, use PC, tablet or smartphone, electronic calculator devices. However, the student may still consult a personal hand written sheet (A4) in which he or she can write useful formulas but not exercises; that sheet must be delivered together with the exam solutions.

To pass the test and gain access to the oral interview you must get a score of at least 16 points (maximum score: 31 points). The oral exam takes place few days before the written exam.

During the oral exam the student will be required to illustrate some aspects of the course topics (definitions, examples, properties, formulas, theorems, proofs, or applications). The logical understanding of concepts, the accuracy and rigor of the mathematical language used to describe them, and the ability to grasp the relationship between abstract aspects and concrete applications will be evaluated.

The final mark is proposed at the end of the oral exam and will take into account all the elements that allow the teacher to evaluate the student's preparation: correctness and completeness in the written test, quality of exposure in the oral interview, active participation during lectures, exercises done during the course.

#### Reference texts

M.Bertsch, R.Dal Passo, L.Giacomelli - "Analisi Matematica" -McGraw-Hill (ISBN978-88-386-6234-8).

The book here above contains all the topics of the course. The books here below are either of Analysis 1 or Analysis 2; our course covers the last topics of Analysis 1 and the first topics of Analysis 2.