Academic Year 2020/2021 - 2° Year

MAT/05 - 9 CFU - 1° Semester

The course aims at conveying to the student the knowledge and comprehensions of the mathematical concepts in the program: sequence and series of functions, limits, derivatives and extrema of functions of several variables, differential equations and systems, Lebesgue theory of integration, curves and differential forms.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

1. **Knowledge and understanding: **The student will learn some concepts of Mathematical

Analysis and will develop both computing ability and the capacity of manipulating some

mathematical structures, as limits, derivatives and integrals for real functions of

more real variables.

2. **Applying knowledge and understanding:** The student will be able to apply the acquired

knowledge in the basic processes of mathematical modeling of classical problems arising from

Engineering.

3. **Making judgements:** The student will be stimulated to autonomously deepen his/her knowledge

and to carry out exercises on the topics covered by the course. Constructive discussion between

students and constant discussion with the teacher will be strongly recommended so that the

student will be able to critically monitor his/her own learning process.

4. **Communication skills: **The frequency of the lessons and the reading of the recommended books

will help the student to be familiar with the rigor of the mathematical language. Through constant

interaction with the teacher, the student will learn to communicate the acquired knowledge with

rigor and clarity, both in oral and written form. At the end of the course the student will have

learned that mathematical language is useful for communicating clearly in the scientific field.

5. **Learning skills:** The student will be guided in the process of perfecting his/her study method.

Inparticular, through suitable guided exercises, he/she will be able to independently tackle new

topics, recognizing the necessary prerequisites to understand them.

The course consists of blackboard lessons on the theoretical parts and subsequent problem sessions. Occasionaly, electronic devices might be used.

Should teaching be carried out in mixed mode or remotely, it may be

necessary to introduce changes with respect to previous statements, in line with the program planned

and outlined in the Syllabus.

Remark: The proofs are not required for th theorems marked with a star (*)

1.Sequences and series of functions. (2 cfu). Real sequences of functions of one real variable. Pointwise and uniform convergence. Characterization of uniform convergence through the suprema sequence. Cauchy test of pointwise and uniform convergence. Limits exchange theorem*, continuity theorem, derivability theorem *, passage of limit under integral sign theorem. Series of real functions of one real variable. Pointwise and uniform convergence. Cauchy test. Absolute and total convergence. Weierstrass test. Comparison among various type of convergence. Theorems of: continuity, derivation and integration by series. Power series. Radius of convergence and related theorem. Cauchy-Hadamard theorem. Abel theorem*. Properties of the sum function of a power series. Taylor series. Conditions for the Taylor expansion. Important expansions (sinus, cosinus, exp, etc.). Fourier series. Sufficient conditions for the Fourier expansion*.

2. FUNCTIONS OF SEVERAL VARIABLES. (2 cfu). Euclidean spaces.Functions between euclidean spaces. Algebra of functions. Composition of functions and inverse function. Limitis of functions . in euclidean spaces. Theorems which characterize the limit by sequences and restrictions. Continuous functions. Continuous functions and connection. Zeros existence theorem. Compactness and continuous functions. Heine-Borel theorem *. Weierstrass theorem. Uniform continuity. Cantor theorem*. Lipschitz functions. Directional and partial derivatives of scalar functions . Differentiable functons. Necessary condtions for differentiability. First derivatives and differential. Derivability of a composition of functions. Higher order derivatives and differentials. Schwartz theorem.*. Second order Taylor formula. al primo e al secondo. Zero gradient theorem. Homogeneous functions and Euler theorem*. Local maximum and minimum for functions of several variables. Fermat theorem . Basic facts about quadratic forms and characterizations of their sign. Second order necessary condition. Second order sufficient conditions. Absolute extremum points search. Basic facts on convex functions. Implicit functions and implicit function theorem (by Dini) for scalar functions of two variables. Scalar and vector implict functions of several variables and related Dini theorems*.

3. DIFFERETIAL EQUATIONS. (2 cfu). First and n order differential equation Systems of n differential equations of first order in n unknown functions. Equivalence between systems and equations. Cauchy problem and definition of its solution. Local and global Cauchy theorem*. Sufficient condition for a function to be Lipschtz. Linear systems. Global solutions of linear systems and structure of the solution set. Wronskian matrix. Lagrange method. Constant coefficients linear systems: construction of a base in the solution space in the case of simple eigenvalues. Linear differential equations of higher order. Euler equation. Solution methods for some specific type of differential equation: separable variable equations, homogeneous equations, Linear equations of the first order. Bernoulli equations.

4. MEASURE AND INTEGRATION. (2 cfu). Basic facts about Lebesgue measure in R^n. Elementary measure of intervals and multi-intervals. Measure of bounded open and closed sets. Measurability for bounded and nonbounded sets. Properties: countable additivity numerabile additività*, monotonicity, upper and lower continuity*, subtractivity . Measurable functions. Basics on the Lebesgue integration theory in R^n: Integration of bounded functions on measurable set of bounded measure. Mean value theorem. Integration of arbitrary measurable functions defined on measurable sets . Geometric meaning of the integral. Integrability tests. Passage of limit under integral sign. Theorem of B.Levi*, Theorem of i Lebesgue*. Integration by series. Method of the invading sets*. Theorem of differentiation under integral sign*. Fubini theorem*. Tonelli theorem*. Reduction formulas for double and triple integrals. Change of variables in integrals*. Polar coordinates in the plane, Spherical and cylindrical coordinates in the space. Comparison between Riemann and Lebesgue integrals*:

5. CURVES AND DIFFERENTIAL FORMS. (1 cfu). Curve in R^n. Simple, plane and Jordan curves. Union of curves. Regular and generally regular curves. Change of parameter. Rectifiable curves. Rectifiabilitry of regular curves*. Curvilinear abscissa. Curvilinear integral. Concept of a differential form and its curvilinear integral. Exact differential forms. Integrability criterion. Circuit integral. Closed forms. Star shaped open sets. Poincaré Theorem *. Simple connected sets. Integrability criterion of simple connected sets *. Regular domains, Green formulas. *.Exact differential equations.

The foreign students who cannot read the italian textbooks can use the following textbook.

Calculus: A Complete Course, 9/E

Robert A. Adams, Christopher Essex, University of Western Ontario

ISBN-10: 0134154363 • ISBN-13: 9780134154367

©2018 • Prentice Hall Canada • Paper, 1168 pp

Published 19 Jun 2017 •

Chapters: 9, 12-13, 14, 18, 11