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First Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)


1. Knowledge and understanding. The student will acquire theoretical and practical knowledge of the topology of the Euclidean spaces, the theory of functions of several real variables, optimization, integral calculus in several variables, the theory of curves and surfaces, and vector fields.
2. Applying knowledge and understanding. The student will be introduced to the main applications of analytical methods to geometric and physical problems.
3. Making judgements. The student will learn how to classify single problems of optimization, calculus of volumes and areas, vector fields in the right class and which method to apply.
4. Communications skills. The course deals, in a somewhat simplified way, with subjects of higher mathematics and aims at providing the student with a rigorous scientific language.
5. Learning skills. Thanks to the notions acquired in this course, the student will be able to address most problems arising in applied science and engineering.


Knowledge of the contents of Matematical analysis 1 and Geometry and algebra is required.


1. Series of numbers and functions (10 hours). Series: definitions of conver gent, divergent, irregular series, Cauchy theorem. Series of positive real numbers: comparison, ratio, root, Raabe, condensation theorems. Series of alternate sign numbers: absolute convergence, Leibniz theorem. Function sequences: pointwise, uniform conve rgence, Cauchy theorem, passage to the limit. Function series: total convergence, power series, radius of convergence. Taylor series, analytic functions.
2. Euclidean spaces (5 hours). Topology of R^n (n=2,3): interior, exterior, boundary, cluster points; open, closed, bounded, compact, connected, compact sets. Polar coordinates.
3. Functions of several real variables (5 hours). Functions defined in R^n: domain, image, graph, level sets. Limit of a functions at a point and at infinity. Continuous functions. Weierstraß, intermediate values theorems. Local, global extrema. Vector - valued functions: limits, continuity.
4. Differential calculus in several variables (15 hours). Partial, directional derivatives. Differentials. Tangent plane to the graph. Higher der ivatives. Hessian matrix. Classification of critical points. Taylor formula. Unconstrained optimization. Constrained optimization (Lagrange multipliers). Vector - valued functions: Jacobian matrix.
5. Integral calculus in several variables (15 hours). Double integrals: reduction formulas, mean value theorem, area of sets in R^2. Triple integrals: reduction formulas, volume of sets in R^3. Variable changes. Rotation volumes.
6. Curves and surfaces (15 hours). Curves in R^n (n=2,3), supports, parameters, Cartes ian and polar equations. Tangent line. (Piecewise) regular, rectifiable curves. Line integrals of the first type. Length of a curve. Jordan curves. Simple, regular surfaces with or without boundary in R^3. Cartesian, parametric equations. Tangent plane. Or ientation of a surface and its boundary. Surface integrals of the first type. Area of a surface. Rotation surfaces.
7. Vector fields (15 hours). Fields in R^n (n=2,3). Divergence, curl, Laplacian. Gradient, irrotational, solenoidal fields. Potential. Line integrals of the second type (circuitation). Surface integrals of the second type (flux). Gauß - Green, divergence, Stokes theorems. Differential forms.

Teaching Methods

Lectures: 50 hours. Exercises: 30 hours. Theory and exercises will be dealt with together, in order to emphasize the close connection between various aspects of the subject. The course will be enriched by tutoring, simulated exams, and a constant support to students. Blackboard, slides, and occasionally calculus software will be employed. The course notes, aimed at complementing the suggested texts, will be made available in due time.

Verification of learning

Verification takes place by means of a written test, consisting of several problems requiring both theoretical and operative knowledge. Such text may be integrated (on request of the student or of the teacher) by an oral examination consisting of some questions. Each part of the exam will be evaluated with a mark between 1 and 30, and the result will be considered positive if the mean of the two marks will be between 18 (sufficient) and 30 (excellent). The “lode” will acknowledge specially brilliant performances. Mean features to be evaluated will be: knowledge of the contents of the course, autonomous elaboration skills, presentation. A further examination mode is available to the students who attend constantly the lectures: it consists of an intermediate written test, to be held at approximatively half of the course duration and dealing with the topics learned up to that moment, which al lows the student to answer on the second half of the program inly at the final examination. The oral examination, if any, still deals with the whole program.


1. C.D. Pagani, S. Salsa, “Analisi Matematica 2”, Zanichelli (2015).
2. S. Salsa, A. Squellati, “Esercizi di Analisi Matematica 2”, Zanichelli (2011).

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