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Subject: CALCULUS 2 (A.A. 2018/2019)


Course year 1
Teaching units Unit analisi matematica II
Mathematics, Information Technology and Statistics (lesson)
  • TAF: Basic compulsory subjects SSD: MAT/05 CFU: 9
Mathematics, Information Technology and Statistics (exercise)
  • TAF: Basic compulsory subjects SSD: MAT/05 CFU: 0
Teachers: Sergio POLIDORO, Luca FERRARI, Stefania PERROTTA
Mandatory prerequisites OFA - Obblighi formativi aggiuntivi
Exam type written
Evaluation final vote
Teaching language Italiano
Contents download pdf download




The course gives a basic knowledge of the differential and integral calculus for functions of several real variables, of the theory of ordinary differential equations.

Admission requirements

Calculus: continuity, differentiability and integrability for functions of one real variable. Sequences and numerical series.

Moreover, the basic knowledge of the linear functions in finite dimension vector space is required.

Course contents

Differential calculus for functions of two (or three) variables: limits and continuity, directional and partial derivatives, differentiability, chain rule. Local extremal points, classification of critical points. Extremal points on curves and surfaces. Lagrange multipliers theorem.

Multiple integrals: double (and triple) integral of a continuous function on a simple domain. Integration methods. Change of variables in a double integral. Polar and spherical coordinates.

Line and surface integrals: general notions about curves. Tangent vectors. Arc length. Line Integral of a vector field. Green’s Theorem Parametric surfaces. Area measure of a surface. Flow integrals. Divergence Theorem and its applications.

Conservative fields: Fundamental Theorem of Calculus for line integrals. Equivalent conditions to the existence of a potential. Simply connected domains. Relation between conservative and irrotational fields.

First order ordinary differential equations: equations with separable variables and linear equations. Cauchy problem: existence and uniqueness of local solutions. Second order linear equations. Characterization of the set of their solutions, explicit solution in the case of constant coefficients.

Teaching methods

Lectures, exercises, tutoring.

Assessment methods

Written and oral exams.

Learning outcomes

1. Knowledge and understanding
Through lectures and individual study, knowledge of the main concepts in Mathematical Analysis, in particular differential and integral calculus for several varibles functions.

2. Applying knowledge and understanding
Through classroom exercises, support activities and individual work, ability to model and solve mathematical problems by analytical tools.

3. Making judgements: Attitude towards a methodological approach that leads to verify claims and methods through rigorous arguments.
Ability to self-assessment of their skills and abilities.

4. Communication skills: Ability to deal with a dialectical discourse in a timely and consistent way, arguing with precision.
Lasting acquisition of mathematical knowledge to be used at any time of their cultural journey.

5. Learning skills: Attitude towards a methodological approach that leads to an improvement in the method of study in order to deepen learning capacity.


N. Fusco, P. Marcellini, C. Sbordone, "Elementi di Analisi Matematica due", Liguori Editore, Napoli.

M.Bramanti, C.D.Pagani, S.Salsa, "Matematica (calcolo infinitesimale e algebra lineare)", Zanichelli, Bologna.

E. Lanconelli, "Analisi Matematica 2", Pitagora, Bologna.