MATHEMATICAL INSTITUTIONS II
Course objectives
The main purpose of the course is to give the student a good knowledge of the basic theory of Differential and Integral Calculus of Functions of several variables, as well as of Linear Algebra. Specific objectives : Knowledge and understanding: at the end of the course the student will have learned the basic theory to study functions of several variables both with scalar or vector values. Moreover he/she will be able to study simple linear maps between vector spaces. Applications : at the end of the course the student will be able to solve simple problems which require the use of the differential and integral calculus . In particular he /she will be able to find extremals of functions of several variables and also determine the potential of a vector field. Moreover he/she will have the ability to solve linear systems. Critical abilities: the student will have the basic knowledge to understand the mathematical tools necessary to study Chemistry and Physics and the way to derive some formulas used in these fields. Communication skills: the student will have the ability to expose the topics studied in written and in oral form. Learning skills: the course will improve the logical abilities and of learning of various scientific disciplines.
Channel 1
VINCENZO NESI
Lecturers' profile
CORRADO MASCIA
Lecturers' profile
Program - Frequency - Exams
Course program
1. Functions of several real variables. Graphic representation. Partial derivatives. Stationary points. Total differential. Properties of the differential. Exact differential. Line integrals. Double integrals. Changes of variables.
2. Functions in three dimensions. Spherical coordinates. Position functions. Volume integrals. The Laplace operator. Other coordinate systems.
3. Vectors. Vectors and their components. Scalar product. Vectorial Product. Scalar fields and vector fields. Gradient of a scalar field. Divergence and rotor of a vector field.
4. Determinants. Determinants of order three. The general case. Linear system solutions. Properties of determinants. Triangular shaped reduction. Alternating functions.
5. Matrices and linear transformations. Special matrices. Matrix algebra. Inverse matrix. Linear transformations. Orthogonal matrices and orthogonal transformations. Symmetries.
6. Eigenvalue matrix problems. Eigenvalue problems. Properties of eigenvectors. Diagonalisation of matrices. Quadratic forms. Complex matrices.
Prerequisites
The course requires basic knowledge such as real numbers, numerical sequences, real functions of real variables.
Books
The instructor will provide notes in Italian, free of charge, which will be made available at the beginning of the course.
Frequency
Participation at lectures is strongly recommended, but not mandatory.
Exam mode
The exam aims to evaluate learning through a written test, which consists in the resolution of problems of the same type as those carried out in classroom exercises and/or assigned as homework, and an oral test discussing the most important topics. relevant topics addressed in the course.
To pass the exam you must achieve a grade of no less than 18/30. To achieve this result, the student must demonstrate that he has acquired a fair amount of manual skill in the use of the concepts introduced in the teaching units of the course, that he is able to carry out at least the simplest of the assigned exercises and that he has understood and assimilated the main definitions and the statements of the main results covered in class.
To achieve a score of 30/30 cum laude, the student must demonstrate that he has acquired excellent familiarity with the calculation tools presented, excellent knowledge of all the topics covered during the course and that he is able to re-elaborate them in a logical and coherent to produce rigorous proofs of small problems or generalizations.
Bibliography
Steiner E.; The Chemistry Maths Book; Oxford University Press
Lesson mode
Lezioni frontali in aula (60%), svolgimento in aula di esercizi (40%)
Channel 2
FLAVIA LANZARA
Lecturers' profile
Program - Frequency - Exams
Course program
Linear Algebra: vector spaces and linear maps, matrices, eigenvalues and eigenvectors, linear systems, quadratic forms and their classifications.
Functions of several variables: limit and continuity, partial derivatives, directional derivatives, differentiability, higher derivatives, Taylor formula, local and global maxima and minima.
Vector valued functions: regular curves and their length, integral on curves, vector fields and their integral on curves. Conservative vector fields.
Integral for functions of several variables : double integrals, volume integrals, reduction and change of variables formulas. Gauss Green and divergence theorems.
Prerequisites
Differential calculus and integrals for functions of one variable.
Books
The instructor will provide notes in Italian, free of charge, which will be made available at the beginning of the course.
Teaching mode
Theoretical lessons (48 hours) and classroom tutorials (36 hours). Weekly home work assignments.
Participation to lectures is recommended, but not mandatory.
Frequency
Participation to lectures is recommended, but not mandatory.
Exam mode
The exam consists of a written test (with problems similar to those seen in class) and an oral exam (about the topics and results seen in class).
The written test can be replaced by two or three intermediate tests carried out during the course that can be taken also by those who do not attend on the sole condition of being subscribed to the e-learning channel of the teacher.
Passing mark is 18/30. The student must prove to have acquired a basic knowledge of the main topics of the course and must be able to solve the simplest exercises assigned.
In order to get the top mark 30/30 cum laude, the student must prove to have acquired an excellent knowledge of all the topics treated during the course, to be able to organize them in a coherent way and to be able to solve all the assigned exercises.
Bibliography
M Bramanti, C.D. Pagani, S. Salsa, "Matematica - Calcolo infinitesimale e algebra lineare" , Zanichelli Editore
L. Lamberti, "Istituzioni di Matematica II " (disponibili on-line sul sito del corso)
For those who wish to explore specific topics:
N. Fusco, P. Marcellini, C. Sbordone, "Lezioni di Analisi Matematica Due", Zanichelli 2020
E. Giusti, "Analisi Matematica 2" – Casa Editrice Bollati Boringhieri 1989
F. Lanzara, E. Montefusco, "Esercizi svolti di Analisi Vettoriale e complementi di teoria".
Lesson mode
Theoretical lessons (24 hours) and classroom tutorials (36 hours). Weekly home work assignments. Participation to lectures is recommended, but not mandatory.
Channel 3
FABIANA LEONI
Lecturers' profile
Program - Frequency - Exams
Course program
Linear Algebra : Vector spaces and linear maps - Matrices - Eigenvalues and eigenvectors - Linear systems - Quadratic forms.
Functions of several variables : Limit and continuity - Partial and directional derivatives and differentiability- Higher derivatives , Taylor formula - Local and global maxima or minima.
Vector valued functions : Regular curves and their length - Integral on curves - Vector fields and their integral on curves - Conservative vector fields.
Integral for functions of several variables : Integrals on planar domains, triple integrals- Reduction and change of variables formulas - Gauss Green and Divergence theorems.
Prerequisites
Basic notions of differential and integral calculus for one variable real functions.
Books
M. Bramanti, C.D. Pagani, S. Salsa "Matematica _ Calcolo infinitesimale e algebra lineare"
Teaching mode
According to the evolution of Covid19 pandemic, the lectures will be frontal and/or on-line.
Frequency
It is advisable to take the classes
Exam mode
The evaluation is based on a written and oral exam. The last one can be granted depending on the result of the written part. The written exam can be substituted by two intermediate partial exams given after the first half of the course and after the end of the course.
Bibliography
Fusco, Marcellini, Sbordone, Elementi di analisi matematica 2. Versione semplificata per i nuovi corsi di laurea, Liguori Ed.
Bramanti, Pagani, Salsa: Analisi Matematica 2. Zanichelli Ed.
Marcellini, Sbordone: Esercitazioni di matematica, vol. 2, parte
I e II. Liguori Ed.
Lesson mode
In room lectures, with use of blackboard
ANTONIO AGRESTI
Lecturers' profile
Channel 4
FILOMENA PACELLA
Lecturers' profile
Program - Frequency - Exams
Course program
Linear Algebra : Vector spaces and linear maps - Matrices - Eigenvalues and eigenvectors - Linear systems - Quadratic forms.
Functions of several variables : Limit and continuity - Partial and directional derivatives and differentiability- Higher derivatives , Taylor formula - Local and global maxima or minima.
Vector valued functions : Regular curves and their length - Integral on curves - Vector fields and their integral on curves - Conservative vector fields.
Integral for functions of several variables : Integrals on planar domains, triple integrals- Recduction and change of variables formulas - Gauss Green and Divergence theorems.
Prerequisites
Differential calculus and integrals for functions of one variable
Books
- Bramanti M., Pagani C.D., Salsa S.: Matematica - Calcolo Infinitesimale e Algebra Lineare - Zanichelli
Teaching mode
The classes of the course will be given in the presence or in hybrid form according to the evolution of the COVID pandemia.
Frequency
It is advisable to take the classes
Exam mode
The evaluation is based on a written and oral exam. The last one can be granted depending on the result of the written part. The written exam can be substituted by two intermediate partial exams given after the first half of the course and after the end of the course.
Bibliography
Marcellini P. - Sbordone C. , Esercitazioni di Matematica II volume - Liguori Editore
Lesson mode
The classes will be given using a blackboard
ANTONIO AGRESTI
Lecturers' profile
- Lesson code1020340
- Academic year2024/2025
- CourseChemical Sciences
- CurriculumSingle curriculum
- Year1st year
- Semester2nd semester
- SSDMAT/05
- CFU6
- Subject areaDiscipline Matematiche, informatiche e fisiche