ANALISI MATEMATICA II

The aim of the course of Mathematical Analysis II is to give the basic skills on the following topics: functions of several real variables, limits and continuity of functions of several real variables, Differential Calculus for real function of several real variables, Integral Calculus for real function of several real variables, curves, surfaces, vectorial fields (resp., differential forms), sequences of functions, series of functions.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

• Knowledge and understanding: The student will learn some basic concepts on functions of several real variables and will develop both computing ability and the capacity of manipulating some common mathematical objects of Mathematical Analysis related to functions of several variables, as limits, partial and directional derivatives of functions of several variables, multiple integrals, vectorial fields (resp., differential forms), sequences of functions, series of functions.
• 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.
• Makin​g 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.
• Communication skills: The attendance 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.
• 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.

Theory lectures and exercises related to the topics covered will be offered. Theory lectures and exercises will be carried out in frontal mode.

For the course of Mathematical Analysis II, 28 hours of theory and 30 hours of other activities (typically, these are exercises) are expected.

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.

In-depth knowledge of the contents of Mathematical Analysis I is indispensable.

Lecture attendance is not compulsory but it is strongly recommended. Active participation in educational activities is extremely beneficial for achieving the learning objectives.

Functions of Several Variables: Introduction, Limits, and Continuity

Topology of Rⁿ. Scalar functions of several real variables. Vector functions of several real variables. Vector fields. Definitions of limits for functions of several real variables. Theorems on limits. Definition of continuous functions. Properties of continuous functions. Weierstrass Theorem. Existence of Zeros Theorem (or Bolzano's Theorem). Intermediate Value Theorem. 

Differential Calculus for Functions of Several Variables

Partial derivatives. Gradient. Directional derivatives. Differentiability. Relationship between differentiability, continuity, and the existence of partial derivatives. Chain rule (derivatives of composite functions). Sufficient condition for differentiability. Geometric interpretation of the partial derivatives of a function at a point. Differentiability and existence of the tangent plane (or hyperplane) to the graph of a function. Higher-order derivatives. Schwarz's Theorem (or Clairaut's Theorem). Hessian matrix. Functions with a vanishing gradient (zero gradient) on a connected set. Taylor's formula. 

Unconstrained Optimization

General concepts of quadratic forms, and definite, semidefinite, and indefinite matrices. Local unconstrained maxima and minima. Fermat's Theorem. Sufficient conditions for unconstrained extrema. 

Constrained Optimization

Implicit functions. Implicit Function Theorem (Dini's Theorem). Derivability of implicit functions. Constrained extrema and the Lagrange Multipliers Theorem. 

Applications.

Least squares regression line equation. Unconstrained optimization in Economics. Production optimization. Consumer problem. 

Multiple Integrals

Introduction. Measure of a set. Measurable sets and negligible sets (sets of measure zero). Multiple integral of a bounded function. Properties of multiple integrals. 

Double Integrals. Integration of a continuous function over a type I (normal with respect to the x-axis) and type II (normal with respect to the y-axis) region. Change of variables theorem in double integrals. Polar coordinates and elliptic coordinates in the plane. 

Triple Integrals. Integration by columns (parallel to a coordinate axis) and by slices (parallel to a coordinate plane). Change of variables theorem in triple integrals. Spherical coordinates (or polar coordinates in space) and cylindrical coordinates in space. 

Applications. Mass, center of mass (centroid), moment of inertia, and volume of a solid of revolution. 

Integrals over Curves and Surfaces

Parametric curves. Definition of a parametric curve, support (trace) of a curve, simple curve, closed curve, smooth (regular) curve, piecewise smooth curve, tangent vector to a curve at a point, orientation induced by a curve on its support, equivalent curves. Properties of equivalent parametric curves. Rectifiable curves, length of a curve, rectifiability theorem for C¹ class curves. Arc length parameter (or curvilinear abscissa). 

Line integrals. Line integral of the first kind (line integral of a continuous real function with respect to arc length). Independence of the line integral of the first kind from the parametrization. Line integral of the second kind (or work integral). Circulation of a vector field along a curve. Dependence of the line integral of a vector field on the orientation induced by the parametrization on the curve. 

Parametric surfaces. Parametric surfaces in R3, definition of a parametric surface, support (trace), simple and smooth (regular) surface, smooth surface patch. Tangent plane to a surface at a point. Normal vector and unit normal vector to a surface at a point. Orientation induced by a surface on its support. Equivalent surfaces. Properties of equivalent surfaces. 

Surface integrals and flux. Surface integral of a continuous real function over a surface. Area of a smooth surface patch. Area of the graph of a function of two real variables. Independence of the surface integral of a function from the surface parametrization. Flux integral (or flux of a continuous vector field through a surface). Dependence of the flux integral of a vector field on the orientation induced by the parametrization on the surface. 

Theorems of Green, Stokes, and Gauss. Green's Theorem, Stokes' Theorem (or Curl Theorem), Gauss' Theorem (or Divergence Theorem), and their applications. 

Power Series

Definition of a power series, radius of convergence. Set of pointwise convergence of a power series. Root Test (Cauchy-Hadamard Theorem). Ratio Test (D'Alembert Theorem). Continuity of the sum of a power series. Term-by-term integration and differentiation theorems for power series. Taylor series expansion. 

[1] C. Canuto, A. Tabacco, Mathematical Analysis 2, Pearson (2023).

The exam of Mathematical Analysis II consists of a written test and of an oral test. Once the written test has been passed, the student will have to take the oral test.

Dates of the exam sessions.

The dates of the exam sessions are available on the degree course website.

Structure of the written test.

Four exercises will be proposed in the written test and it lasts 120 minutes.

Evaluation of the written test.

The maximum score obtainable in the written test is 30/30. The written test is considered passed if the student has achieved a score of at least 18/30. A score will be assigned to each exercise. Each exercise will be assigned the maximum score expected if and only if it is carried out correctly. Otherwise, a partial score will be assigned that will be determined based on the errors made. In the event that the total score is greater than or equal to 15 and less than 18, the Examination Commission may admit the student to the oral exam with reservations and may require some preliminary exercises to the student.

Oral exam.

The oral exam covers all the contents of the course (see the section “Detailed Course Contents” in the Syllabus related to Module A and in the Syllabus related to Module B). The final evaluation will take into account the outcome of the written exam. The schedule of the oral tests will be prepared by the Examination Commission. If the student does not pass the oral exam or decides not to show up for the convocation, it will be necessary to take the exam again according to Mode 2.

Grading criteria.

Written and oral tests will assess students' understanding of the topics covered in the course and their ability to use the relevant language. A successful exam requires a complete and accurate presentation of definitions, statements, and examples. Assessment of content acquisition also includes theorem proofs, where applicable. The final grade is expressed in thirtieths according to the following table:

• NOT PASSED (<18): The student demonstrates a poor and fragmented knowledge of the subject matter, exhibits serious comprehension errors and do not present the contents in an acceptable manner;
• 18-21: The student demonstrates limited knowledge and a basic understanding of the subject matter, presents the contents unclearly and with little precision;
• 22-24: The student demonstrates an acceptable knowledge and a basic understanding of the subject matter, presents the contents correctly but not in a fully structured way;
• 25-27: The student demonstrates a broad knowledge and adequate understanding of the subject matter, presents the contents correctly but not in a complete way;
• 28-29: The student demonstrates an in-depth knowledge and a solid understanding of the subject matter, presents the contents clearly and in a fully structured way;
• 30-30 cum laude: The student demonstrates a complete and detailed knowledge and an excellent understanding of the subject matter, presents the contents clearly and in a fully structured way.

Note. Information for students with disabilities and / or SLD.

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory

and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA). It is also possible to contact the referent teacher CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department of Electrical, Electronic and Computer Engineering.

Note. Learning assessment may also be carried out on-line, should the conditions require it.

All the topics mentioned in the program can be requested during the exam.

The attendance of the lessons, the study of the recommended texts and the study of the material provided by the teacher (handouts and collections of exercises carried out and proposed) allow the student to have a clear and detailed idea of ​​the questions that may be proposed during the exam.

An adequate exposition of the theory involves the use of the rigorous language characteristic of the discipline, the exposition of simple examples and counterexamples that clarify the concepts exposed (definitions, propositions, theorems, corollaries).

The main types of exercises related to the Module A of the course of Mathematical Analysis I are:

●      Limits of real functions of several real variables, analysis of the continuity, existence of partial/directional derivatives, analysis of the differentiability of real functions of several real variables.

●      Finding the free and constrained extrema of real functions of several real variables.

●      Finding the global extrema of real functions of several real variables.

●      Calculation of double integrals and triple integrals.

●      Calculation of line integrals.

●      Calculation of surface integrals and flux integrals.

●      Study of the conservative vector fields and finding of the potentials of conservative vector fields. Study of exact differential forms and finding of the primitives of and exact differential form.

●      Calculation of line integrals of vector fields (resp., differential forms).

●      Study of the convergence of a power series.