Mathematical Analysis II 1

Module ESERCITAZIONI

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
Physics.
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.

1.Sequences and series of functions.  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.  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. 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. 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. 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.

6. SURFACES AND SURFACE INTEGRALS. Regular surfaces. Tangent plane and normal unit vector. Area of a surface. Surface integrals. Stokes formula and divergence theorem.

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

The Mathematical Analysis II exam can be passed in two ways.

Mode A: written test and subsequent oral test

In this mode, only one written test is proposed. Once the written test has been passed, the student will have to take the oral test. The written test lasts 120 minutes.

Structure of the written test.

In the written test two definitions and four exercises will be proposed.

Evaluation of the written test.

The maximum score that can be obtained in the written test is 30/30. The written test is considered passed if the

student has achieved a score of at least 18/30. The student get 18/30 if and only if

correctly provides one of the two proposed definitions and correctly solves two of the

four proposed exercises.

Oral test and final score.

The oral exam covers all the topics of the course. The final score takes into account the  score obtained in the written test and the evaluation of the oral test.

Mode B: ongoing written tests and subsequent oral test

In this mode there are two ongoing written tests: the first test 

at the end of the first teaching period and the second test at the end of the second teaching period. Once both have been passed, an oral test is scheduled. 

Structure of ongoing tests.

Each ongoing test has the same structure. In each test two definitions and four exercises will be proposed.

Evaluation of ongoing tests.

The maximum score that can be obtained in each ongoing test is 30/30. Each ongoing test is considered

passed if the student has achieved a score of at least 18/30. The student get 18/30 if and only if correctly provides one of the two proposed definitions and correctly solves two of the four proposed exercises.

Oral test and final score.

The oral exam covers all the topics of the course. The final score takes into account the  score obtained in the outgoing tests and the evaluation of the oral test.

Radius of convergence for power series.

 Existence of zeroes Theorem.

Measurable functions.

Linear differential equations.

The students will be able to find examples of exercises on Studium.