#
Mathematical methods applied to chemistry

A.Y. 2021/2022

Learning objectives

The course is intended to give a good knowledge of differential equations, multiple integration,potential theory and some of their connections.

Expected learning outcomes

Students will master concepts and computational techniques that are useful in the study of the solutions of some differential equations involved in mathematical models that describe some natural phenomena.

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Most of the material will be presented in remote-mode, partly using the delayed-time mode (video-lessons will be posted on the Ariel platform); and partly using the real-time mode (problem solving, q&a, discussions will be held using the Zoom platform).

This choice is aimed to let the students free to attend the lab activities.

Hopefully, some frontal teaching will be possible.

The usual rules for the exams may be modified, possibly devoting more space to the oral part of the exam.

This choice is aimed to let the students free to attend the lab activities.

Hopefully, some frontal teaching will be possible.

The usual rules for the exams may be modified, possibly devoting more space to the oral part of the exam.

**Course syllabus**

1] Differential calculus for functions of several real variables. Curves p: R¹→Rᵐ: orientation, arc-length , I-type line integrals and applications.

Functions f:Rⁿ→R: continuity, differenziability, second derivatives, Taylor's formula at II order.

Maps F:Rⁿ→Rᵐ: jacobian matrix, composition. Applications in R^3: polar and spherical coordinates. Surfaces. Free optimization: stationary points, hessian matrix, quadratic forms, the eigenvalues test. Constrained optimization: implicit functions, Dini's theorem. The Lagrange multipliers method.

2] Integral Calculus in 2 and 3 variables The basic properties of the Riemann integral for f:R²→R; iterated integrals for regular sets, measurable functions, sets of measure zero. Change of variables. Improper double integrals. The Riemann integral for f:Rᶟ→R: integration methods ("lines" or "layers"). I type surface integrals.

3] Vector fields F:Rᶟ→Rᶟ: the differential operators grad, div, rot and their properties. Line integrals of II type: the work of a vector field along a line. Conservative and irrotational vector fields, potentials, Poincaré`s lemma, simply connected sets. Surface integrals of II type. The flow of a vector field. Surfaces with boundary. Integral formulas: Gauss-Green's formula, the Divergence Theorem in dimensions 2 and 3, Stokes's theorem.

4] Differential equations. Ordinary differential equations: Cauchy's problems, exixtence/uniqueness of local/global solutions. Types of equations: separable, linear of order 1, Bernoulli, linear of order 2. Lagrange's method. Constant coefficients linear equations. Harmonic oscillators. Few hints about PDEs.

5] Depending on the amount of time left, an extra argument might be included in the programme. The content of this extra argument may vary from year to year, and it depends on the interest of the students.

Functions f:Rⁿ→R: continuity, differenziability, second derivatives, Taylor's formula at II order.

Maps F:Rⁿ→Rᵐ: jacobian matrix, composition. Applications in R^3: polar and spherical coordinates. Surfaces. Free optimization: stationary points, hessian matrix, quadratic forms, the eigenvalues test. Constrained optimization: implicit functions, Dini's theorem. The Lagrange multipliers method.

2] Integral Calculus in 2 and 3 variables The basic properties of the Riemann integral for f:R²→R; iterated integrals for regular sets, measurable functions, sets of measure zero. Change of variables. Improper double integrals. The Riemann integral for f:Rᶟ→R: integration methods ("lines" or "layers"). I type surface integrals.

3] Vector fields F:Rᶟ→Rᶟ: the differential operators grad, div, rot and their properties. Line integrals of II type: the work of a vector field along a line. Conservative and irrotational vector fields, potentials, Poincaré`s lemma, simply connected sets. Surface integrals of II type. The flow of a vector field. Surfaces with boundary. Integral formulas: Gauss-Green's formula, the Divergence Theorem in dimensions 2 and 3, Stokes's theorem.

4] Differential equations. Ordinary differential equations: Cauchy's problems, exixtence/uniqueness of local/global solutions. Types of equations: separable, linear of order 1, Bernoulli, linear of order 2. Lagrange's method. Constant coefficients linear equations. Harmonic oscillators. Few hints about PDEs.

5] Depending on the amount of time left, an extra argument might be included in the programme. The content of this extra argument may vary from year to year, and it depends on the interest of the students.

**Prerequisites for admission**

It is strongly suggested that the student has already passed the exam "Istituzioni di Matematica".

**Teaching methods**

When the teaching activity has no limitations:

Frontal teaching. Problem sessions. Homeworks are assigned, and their solution is discussed within the problem sessions.

Frontal teaching. Problem sessions. Homeworks are assigned, and their solution is discussed within the problem sessions.

**Teaching Resources**

- M. Bramanti, C. Pagani, S. Salsa, Analisi Matematica 2, Zanichelli ed.;

- G. Turrell, Mathematics for Chemistry and Physics, AcademicPress, 2002.

- other possible notes written by the teacher.

- G. Turrell, Mathematics for Chemistry and Physics, AcademicPress, 2002.

- other possible notes written by the teacher.

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student is required to solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in Mathematical Analysis. The duration of the written exam is proportional to the number of exercises assigned, taking into account the nature and complexity of the exercises themselves (however, the duration does not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams (this second option may not be available in case of remote-teaching modality). The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student is required to illustrate results presented during the course and may be required to solve problems regarding Mathematical Analysis, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and are communicated immediately after the oral examination.

- During the written exam, the student is required to solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in Mathematical Analysis. The duration of the written exam is proportional to the number of exercises assigned, taking into account the nature and complexity of the exercises themselves (however, the duration does not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams (this second option may not be available in case of remote-teaching modality). The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student is required to illustrate results presented during the course and may be required to solve problems regarding Mathematical Analysis, in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and are communicated immediately after the oral examination.

MAT/01 - MATHEMATICAL LOGIC

MAT/02 - ALGEBRA

MAT/03 - GEOMETRY

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS

MAT/05 - MATHEMATICAL ANALYSIS

MAT/06 - PROBABILITY AND STATISTICS

MAT/07 - MATHEMATICAL PHYSICS

MAT/08 - NUMERICAL ANALYSIS

MAT/09 - OPERATIONS RESEARCH

MAT/02 - ALGEBRA

MAT/03 - GEOMETRY

MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS

MAT/05 - MATHEMATICAL ANALYSIS

MAT/06 - PROBABILITY AND STATISTICS

MAT/07 - MATHEMATICAL PHYSICS

MAT/08 - NUMERICAL ANALYSIS

MAT/09 - OPERATIONS RESEARCH

Practicals: 16 hours

Lessons: 40 hours

Lessons: 40 hours

Professors:
Vesely Libor, Vignati Marco

Educational website(s)

Professor(s)

Reception:

Wed, 12.30 am - 2.00 pm; otherwise, contact me via e-mail

Math Dept., via C.Saldini 50, room R013, ground floor