Teachings
IN/0188  MATHEMATICAL ANALYSIS 2
Academic Year 2022/2023
Free text for the University
 Professor

FRANCESCO ARRAI (Tit.)
 Period

First Semester
 Teaching style

Convenzionale
 Lingua Insegnamento

ITALIANO
Informazioni aggiuntive
Course  Curriculum  CFU  Length(h) 

[70/72] CIVIL ENGINEERING  [72/00  Ord. 2021] PERCORSO COMUNE  8  80 
[70/73] ENVIRONMENTAL AND LAND ENGINEERING  [73/00  Ord. 2020] PERCORSO COMUNE  8  80 
[70/77] CHEMICAL ENGINEERING  [77/00  Ord. 2020] PERCORSO COMUNE  8  80 
Objectives
1. Knowledge and understanding. At the end of the course the student will have knowledge of topics concerning real sequences, the infinitesimal calculus of real functions of a real variable.
2. Applying knowledge and understanding. The student will be introduced to the main applications of the theoretical notions of the course, concerning both the solution of mathematical problems and the study of physical problems.
3. Autonomy of judgment. The student will learn how to classify single problems of differential and integral calculus.
4. Communicative Skills. Students will acquire the ability to communicate what they learn and elaborate and also to express and argue the choice of methodology with respect to another to solve a mathematical problem.
5. Learning skills. Thanks to the notions acquired in this course, the student will be able to perfect his knowledge of higher mathematics and its applications to engineering.
Objectives
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.
Prerequisites
Knowledge of the contents of Matematical analysis 1 and Geometry and algebra is required.
Prerequisites
Good knowledge of algebra, trigonometry and elementary analytic geometry.
Contents
1. Series of numbers and functions (10 hours). Series: definitions of convergent, divergent, irregular series, Cauchy theorem. Series of positive real numbers: comparison, ratio, root, Series of alternate sign numbers: absolute convergence, Leibniz theorem. Function sequences: pointwise, uniform convergence, 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. Vectorvalued functions: limits, continuity. 4. Differential calculus in several variables (15 hours). Partial, directional derivatives. Differentials. Tangent plane to the graph. Higher derivatives. Hessian matrix. Classification of critical points. Taylor formula. Unconstrained optimization. Constrained optimization (Lagrange multipliers). Vectorvalued 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, Cartesian 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. Orientation 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). GaußGreen, divergence, Stokes theorems. Differential forms.
Contents
Functions in R^N . Sets in R^N : accumulation, isolated, internal , external, boundary points; bounded, open, closed, compact, connected sets.Functions of several variables in RN , domain, image; definition and properties of limits. Limit theorems. Continuity. Partial derivatives. Higher derivatives. The total differential. Tangent plane. Taylor formula. The chain rule. Implicit functions.
Maxima and minima.
Local maximum and minimum. Critical points of a smooth function. The second derivative test. Maximum principle for harmonic functions. Maxima and minima with constraints.
Curves and surfaces.
Plane Curves: parametric equations, implicit form, smooth, piecewise, simple, closed, oriented curves.Length of arc in parametric, Cartesian, polar form. Curves in R3 .Surfaces in R3: parametric and Cartesian form, normal vector to a surface.
Multiple integrals.
Double integral. Integration over normal regions. Normal regions in polar coordinates. Triple integral: Integration over normal regions, cylindrical coordinates and applications. Volume.
Line and surface integrals. Line integrals, applications. Area of smooth surfaces, of a surface of revolution. Surfaces Integrals. Flux.
Green’s theorem, Stokes’ theorem and their applications.
 Sequences of functions, Series of real numbers and functions. Convergence.
Comparison tests. Positive series. Alternating series.
Contents
1. Series of numbers and functions (10 hours). Series: definitions of convergent, divergent, irregular series, Cauchy theorem. Series of positive real numbers: comparison, ratio, root, Series of alternate sign numbers: absolute convergence, Leibniz theorem. Function sequences: pointwise, uniform convergence, 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. Vectorvalued functions: limits, continuity. 4. Differential calculus in several variables (15 hours). Partial, directional derivatives. Differentials. Tangent plane to the graph. Higher derivatives. Hessian matrix. Classification of critical points. Taylor formula. Unconstrained optimization. Constrained optimization (Lagrange multipliers). Vectorvalued 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, Cartesian 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. Orientation 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). GaußGreen, divergence, Stokes theorems. Differential forms.
Teaching Methods
Frontal lectures (theory): 50 hours
Frontal lectures (exercises): 30 hours
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
The verification test consists of a written test. The written test consists of 6 exercises related to optimization, integration, differentiation, series.
Texts
 Analisi matematica II (teoria ed esercizi).
casa editrice Springer.
autori Claudio Canuto, Anita Tabacco.
Analisi Matematica 2.
casa editrice Zanichelli,
autori Paolo Marcellini, Carlo Sbordone.
 Esercitazioni di Matematica Due. Prima e Seconda parte.
casa editrice Zanichelli.
autori Paolo Marcellini, Carlo Sbordone.
More Information
Exercises shared on the telegram channel
More Information
Exercises shared on the telegram channel