Salta ai contenuti. | Salta alla navigazione

Strumenti personali


Academic year and teacher
If you can't find the course description that you're looking for in the above list, please see the following instructions >>
Versione italiana
Academic year
Didactic period
Primo Semestre

Training objectives

Aim of the course is to provide the basic tools of differential and integrals calculus for functions with several real variables; the case of vector valued functions is also considered. Ordinary differential equations are also studied.

The main knowledge that the student has to reach shall be that of recognize the typology of the problems; for instante in the case of differenzial equation he has to recognize the typology of the equation, in differential calculus he has to distinguish among implicit and parametrised manifolds and change of variables,
in the integral calculus he has to distinguish among multiple, linear and area integrals, and in the case of sequence and Series of functions their typology, in such a way that the adequate resolution method can be applied.

The main skill to be achieved will be the followoings; ability to model a problem by its mathematical formulation and ability to solve the described mathematical model, both through full analytical solution, or through the use of numerical tools.


A course in differential and integral calculus. Such course must develop the concepts of limit, derivative and integral for real functions of a real variable. Elementary matrix theory. There are then considered necessary
the knowledges gained during the courses of Calculus 1 and Geometry.

Course programme

Preliminaries: normed spaces and Euclidean spaces. Distance and norm and definition of limit and continuity for functions between Euclidean spaces. (2 hours)

Parameterized and regular curves. Length of a curve and length of a smooth curve. Kinematic decomposition of the acceleration. (10 hours)

Elements of topology. Continuous functions and their characterization by sequences. Closed and open sets via continuous functions and characterization of closed sets by successions.
Study of the continuity of functions, through restrictions on continuous curves and using polar coordinates.
Path connected sets, bounded sets, compact sets. Theorem of existence of zeroes for continuous functions on path connected sets and Weierstrass theorem on compact sets. Level sets and determination of maximum and minimum through the level sets. (6 hours)

Differential calculus; partial derivatives, differential and Theorem of the total differential. Tangent plane, normal line, directional derivatives and derivation formulas for composite functions. Maximum and minimum growth and the relationship between the gradient and level set of a function; Implicit function theorem (Dini). (14 hours)

Regular parameterized surfaces; tangent plane, normal line. Implicit function theorem in the general formulation.
Landau symbol and higher order Taylor expansion. Applications of the theory of smooth surfaces and curves in the plane and in space. Surfaces of rotation; sphere, cone, cylinder, torus, and ellipsoid. (8 hours)

Maximum and minimum for several variables functions. Free extremal points; necessary conditions in the case of differentiable functions and free stationary points. Sufficient conditions for the extremality of stationary points; Taylor's formula of the second order. Maximum and minimum of compact sets; parameterization method, replacing the constraint and Lagrange multipliers. (12 hours)

Diffeomorphisms and local invertibility theorem. Change of variables; linear maps, polar, cylindrical and spherical coordinates. (8 hours)

Integration theory; line integrals, definition of the multiple integral on rectangles; Riemann integration, through the definition of upper and lower integral and definition of integrable function. Extension of the Integral on bounded domains; definition of measurable set, and necessary and sufficient conditions for measurability. Definition of simple set in the plan and reduction formula for the calculation of the double integral. Change of variables in the computation of double integrals. Definition of absolutely integrable function in a generalized sense. Simple and stratified domains in space. Computation of the volumes for regular solids and for solids of rotation. Change of variable formula in triple integrals. (18 hours)

Computation of the area of ¿¿a parameterized surface and formulas for the calculation of the area of ¿¿the Cartesian and rotation surfaces; surface integral. Application of surface integrals in the study of vector fields; conservative fields, sufficient conditions for a field to be conservative. Divergence and Stokes theorems and their applications in the study of conservative fields and the calculation of flows through surfaces. (12 hours)

Sequences of functions; pointwise and uniform convergence and their properties. Series of functions; pointwise, absolute pointwise convergence, uniform, absolute and total convergence and their properties. Power and Taylor series. Fourier series; property and Parseval formula. (16 hours)

Differential equations; Cauchy problem, existence and uniqueness theorem.
Equations of the first and second order; mechanical vibrations. (14 hours)

Didactic methods

The course is organized in theoretical and practical lectures. Theoretical lectures are held in the classroom and are thought to present the arguments included in the program, trying to keep a level of compromise between a rigorous mathematical exposition (hence trying to give, at least partially, the proof of the theorems) and an applicative level (hence by giving and solve problems and exercises). The practical lectures are divided in two different activities. A first one is held in classroom and is thought to give and solve exercises to better explain the theoretical part and to give ideas of the applications.

The lessons shall use GoogleClassRoom, code


Learning assessment procedures

The exam of Calculus 2 is divided in two different steps, to be done in two different days; at first there is a written exam in which the candidate is required to solve five exercises with a given time of 3 hours, the second step is an oral exam in which the knowledges of the candidate on a theoretical basis are investigated by asking three different questions.

The written exam is considered valid if the evaluation is bigger or equal to
15/30 and the possible arguments of the exercises can vary among the following possibilities:

1. (6 points) Differential equations; first order differential equations (separable variables, linear, homogeneous and of Bernoulli type) and second order differential equations (the ones that can be reduced to first order, autonomous, linear with constant coefficients):
2. (6 points) Differential calculus; determination of tangent and normal lines and planes to surfaces and curves, defined both in the parametric and in implicit form.
3. (6 points) Maximum and minimum problems; determination of maximum and minimum, local and absolute on compact and non compact sets, classification of free stationary points.
4. (6 points) Integration theory; line integrals, double, triple and surface integrals, with applications.
5. (6 points) Sequences and series of functions; power series, Taylor series and Fourier series.

The written exam is considered valid if the evaluation is bigger or equal to
15/30 and remains valid within the same session; so, if one pass the written exam in december, january or february, there is time till to the beginning of march to give the oral exam (but not in june), and if one pass the written exam in june, july or september, there is time till to september to give the oral exam (but not in december).
The candidate that pass the first step can access to the second step, in which he/her is asked to answer to three question on a theoretical basis;
the candidate is asked to give an exposition as much exhaustive as possible of the arguments. It is up to the candidate to chose the level of deepening of such exposition, including the presentation of suitable examples, counterexamples and proofs of the cited Theorems. In principle, a first question can be done starting on the written exam, the second two questions can be done among all the other arguments belonging to the program of the course.

The final evaluation takes into account the results of the written exam and of the level of the oral exam. It is not necessary the average of the two evaluations. In case the oral exam is not sufficient, depending on the level, it is up to the teacher to ask the candidate to repeat both steps or only the oral exam.
During the exams it is forbidden the use of texts and any electronic device,
cell phones included.

Reference texts

As refence book we advice the following;

M. Bramanti, C. D. Pagani, S. Salsa:
"Analisi Matematica 2",
Zanichelli, 2009.

To deepen the knowledge, we sugest the following books:

S. Salsa, A. Squellati:
Esercizi di Matematica II, Zanichelli, 2002.

B. Demidovic:
Esercizi e problemi di Analisi Matematica,
Editori Riuniti, 1999.

Beside the texts in the previous list, some notes written by the teacher are distributed to the students. In details, on the website of the course the following material can be found:
- Text and solutions of the exams of Calculus 2 given in the past years;
- Notes for the MatLab Laboratory;
- Exercises with solutions relative to all the arguments treated during the course;
- Notes of a part of the course that the teacher considers not covered by the texts given as reference books.