Mathematical Analysis II M - Z

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.

In particular, 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. 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. DIFFERENTIAL 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