Aim of the course is to give to the students some basic tools and topics in mathematical analysis, for functions both with two or several variables.

01. Differential calculus for functions of several variables:

01.01  Preliminaries.

01.02  Domain, graph and level curves.

01.03  Topology in Rn: distance and its properties, neighborhood, open and closed sets and their properties.

01.04  Limits.

01.05  Calculus of limits: restrictions method and polar coordinates.

01.06  Continuity.

01.07  Weierstrass Theorem.

01.08  Partial derivatives and gradient.

01.09  Geometric meaning of partial derivatives.

01.10  Directional derivatives.

01.11  Tangent plane.

01.12  Differentiability and linear approximation.

01.13  C^1 functions are differentiable*.

01.14  Relation between directional derivatives and gradient of a differentiable function.

01.15  Algebra of derivatives.

01.16  Chain rule.

01.17  Higher-order derivatives.

01.18  Schwarz Theorem*.

01.19  Taylor’s formula in R2.

02. Curves in Rn:

02.01  Vector-valued functions.

02.02  Limits and continuity.

02.03  Derivatives for vector-valued functions.

02.04  Chain rule for vector-valued functions.

02.05  Curves in R2 and in Rn.

02.06  Parametric representation of a curve.

02.07  Closed curves and simple curves.  

02.08  Curves in the plane and in the space.

02.09  Parametrization of curves in the plane: lines and line segments.

02.10  Parametric equations of conics.

02.11  Graphs of functions.

02.12  Parametrizations of curves in the space.

02.13  Smooth curves.

02.14  Tangent vector.

02.15  Piecewise smooth curves.

03. Optimization:

03.01  Maxima and minima for functions of one or several variables.

03.02  Critical points.

03.03  First order necessary condition (Fermat Theorem).

03.04  Hessian matrix.

03.05  Classification of critical points in R2 and in Rn.

03.06  Constrained maxima and minima of a function.

03.07  The method of Lagrange multipliers.

04. Integral calculus for functions of two variables:

04.01  Double integrals over a rectangle.

04.02  Iteration formula over a rectangle.

04.03  Geometric meaning of double integrals.

04.04  Double integrals over more general domains.

04.05  Properties of the double integral.

04.06  Iteration of double integrals in Cartesian coordinates.  

04.07  Change of variables in R2.

04.08  Change of variables in double integrals.

04.09  Improper double integrals.

05. Ordinary differential equations:

05.01  Preliminaries.

05.02  Separable equations.

05.03  Linear differential equations: preliminaries and superposition principle.

05.04  First-order linear equations.

05.05  Homogeneous constant-coefficients linear equations of higher-order: solution.

05.06  Non-homogeneous constant-coefficients linear equations of higher-order: solution.

05.07  Euler equation and Bernoulli equation.

05.08  Non-linear differential equations. 

05.09  Cauchy problem.

05.10  Local existence and uniqueness for the Cauchy problem.

05.11  Boundary value problems. 

* : this means that the proof is required.

There are no mandatory prerequisites. It is recommended to take the exam of Calculus 2 during the second year of the Laurea Degree Program in Applied Computer Science and after Calculus 1.

Knowledge and understanding:

The student will learn the basic notions of mathematical analysis for the study of functions of several variables.

Applying knowledge and understanding:

The student will learn the methodologies of mathematical analysis and will be able to apply them to the study of various problems.

Making judgements:

The student will be able to apply the techniques of mathematical analysis in order to solve new problems, also coming from real-world applications.

Communications skills:

The student will have the ability to express the fundamental notions of mathematical analysis using a rigorous terminology.

Learning skills:

The student will learn the ability to study the notions of mathematical analysis, also in order to use it in solving different kind of problems.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Adams, Calcolo Differenziale 2, Casa Editrice Ambrosiana

Adams - Essex, Calculus: a complete course, Pearson Canada

Barutello - Conti - Ferrario - Terracini - Verzini, Analisi matematica, Vol.2, Apogeo

Bramanti - Pagani - Salsa, Analisi matematica 2, Zanichelli

Salsa - Squellati, Esercizi di Analisi matematica 2, Zanichelli

Assessment

The exam of Calculus 2 consists of a written exam and an oral one, both of them mandatory.

The written exam, to carry out in two hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.

The oral exam consists of a discussion related to the topics of the course.  The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

The final mark of Calculus 2 is the average of the marks of the written exam and the oral one.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Adams, Calcolo Differenziale 2, Casa Editrice Ambrosiana

Adams - Essex, Calculus: a complete course, Pearson Canada

Barutello - Conti - Ferrario - Terracini - Verzini, Analisi matematica, Vol.2, Apogeo

Bramanti - Pagani - Salsa, Analisi matematica 2, Zanichelli

Salsa - Squellati, Esercizi di Analisi matematica 2, Zanichelli

Assessment

The exam of Calculus 2 consists of a written exam and an oral one, both of them mandatory.

The written exam, to carry out in two hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.

The oral exam consists of a discussion related to the topics of the course.  The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

The final mark of Calculus 2 is the average of the marks of the written exam and the oral one.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.