MATHEMATICAL ANALYSIS II
Course objectives
The student will be able to study and use - curves and surfaces - partial derivatives and directional functions of several variables - two-dimensional and three-dimensional domains, - curvilinear coordinates (polar, spherical, cylindrical), - multiple integrals, surface integrals and line integrals of functions, - exactness of differential forms and their potentials, - line integrals over simple and closed circuits, - differential operators, flows, Divergence Theorem and Stokes’ Theorem, - power series, Taylor series, Fourier series. SPECIFIC OBJECTIVES KNOWLEDGE AND UNDERSTANDING. The course will allow an in-depth comprehension of the fundamental concepts and tools of the Analysis of functions of several variables; in particular: the differentiation and integration of function of several variables; curves and surfaces; differential opertators: gradient, divergence, curl, laplacian; sequences and series of functions, with particular attention on Taylor and Fourier series. CAPABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING. Through this course students will be able to apply the aforementioned tools mainly to Physics and Engineering problems: partial differential equations, vector fields, centers of mass, moments of inertia, work and potentials, conservative and non-conservative forces, Gauss Theorem, Stokes Theorem, Maxwell Equations. MAKING AUTONOMOUS JUDGEMENTS. After the course, students will be able to choose, for a given Physics or Engineering problem, the best methodology, through the deep understanding of the requirements and constraints of the context. COMMUNICATE SKILLS. At the end of the course the students will be able to illustrate the importance of the tools studied in the course, with the aim of applying them to Physics and Engineering problems, such as the signal reconstruction, the study of typical problems in Fluidodymìnamics, Electromagnetism, Hydrodynamics and in general problems which imply the use of the differential and integral calculus for functions of several variables. LEARNING SKILLS. The student will develop the capability to autonomously study the theoretical topics characterizing the course and their application to practical problems in Physycs and Engineering.
Channel 1
ALBERTO MARIA BERSANI
Lecturers' profile
Program - Frequency - Exams
Course program
Curves. Review of vector calculus. Vector valued functions, limits and
continuity. Regular curves and differential vector calculus. Length of a curve arc. arc parameter or
arc length. Elements of differential geometry of curves (outline): tangent, normal, curvature,
twist, intrinsic triad (10 hours).
Differential calculus for real functions of several variables. Graphics and level sets. Limits and continuous
functions of several variables. Topology in Rn and properties of continuous functions (10 hours).
Partial derivatives, tangent plane, differential, directional derivatives. Higher order derivatives and successive approximations.
Partial differential equations and classification of second-order equations (introduction) (10 hours).
Unconstrained optimization (5 hours).
Differential calculus for vector-valued functions of several variables. Vector-valued functions of several variables: generalities. Surfaces in parametric form. Limits, continuity and differentiability for functions f: Rn → Rm. Smooth surfaces in parametric form. Coordinate transformations and their inverses (10 hours).
Constrained optimization (5 hours).
Integral calculus for functions of several variables. Double integrals. The calculation of triple integrals (10 hours).
Vector fields. Field lines. Gradient, divergence and curl. Differential forms and
work. Line integrals of the second kind. Irrotational, solenoidal, conservative fields. Potential.
Gauss-Green formula in the plane. Area and surface integrals. Surface integral of a vector field (flux). Divergence (or Gauss) Theorem. Curl (or Stokes) Theorem (10 hours).
Sequences of functions. Pointwise and uniform convergence (10 hours).
Series of functions and total convergence. Power series and Taylor series. Trigonometric series and
Fourier series. Pointwise and total convergence of Fourier series (10 hours).
Prerequisites
Geometry and Mathematical Analysis 1: elements of Linear Algebra (vectors, matrixes, and operations); elements of Plane and Space Geometry (lines, planes, conics, quadrics); elements of theory of linear transformations; Elements of Mathematical Analysis (complex numbers, sequences, series, limits, derivatives, functions and graphs, Taylor polynomials, definite, indefinite and improper integrals, ordinary differential equation).
Books
M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 2. Zanichelli, 2009.
M. Amar, A.M. Bersani: ANALISI MATEMATICA I – Esercizi e richiami di teoria. Amazon 2022.
M. Amar, A.M. Bersani: ANALISI MATEMATICA II – Esercizi e richiami di teoria. Amazon 2022.
D. Andreucci, A.M. Bersani: RISOLUZIONI DI PROBLEMI D’ESAME DI ANALISI MATEMATICA II. Esculapio, 1998.
online supplementary educational materials on the web page
http://www.dmmm.uniroma1.it/~alberto.bersani/A2.htm
Teaching mode
The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.
Frequency
The attendance is optional, but strongly recommended
Exam mode
At the end of the course, starting from June, the written exam (lasting 2 hours and half) consists of 5 exercises concerning the topics of the programme, with the aim of testing the knowledge level of the several techniques useful to solve the problems of the final test.
If the written exam receives an evaluation greater or equal to 15/30, the theory exam (written - 1 hour and half - or oral - 1 hour -, depending on student's request), consists of 2 theory questions (definitions, theorem statements, proofs, examples, counterexamples), starting from the results and mistakes in the written exam, in order to test the knowledge level of the theoretical aspects of the course.
The final score is a weighted mean between the scores of the written exam and of the oral exam. In both the tests the student must obtain a score greater or equal to 15/30.
Bibliography
M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 2. Zanichelli, 2009.
M. Amar, A.M. Bersani: ANALISI MATEMATICA I – Esercizi e richiami di teoria. Amazon 2022.
M. Amar, A.M. Bersani: ANALISI MATEMATICA II – Esercizi e richiami di teoria. Amazon 2022.
D. Andreucci, A.M. Bersani: RISOLUZIONI DI PROBLEMI D’ESAME DI ANALISI MATEMATICA II. Esculapio, 1998.
online supplementary educational materials on the web page
http://www.dmmm.uniroma1.it/~alberto.bersani/A2.htm
Lesson mode
The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.
ALBERTO MARIA BERSANI
Lecturers' profile
Program - Frequency - Exams
Course program
Curves. Review of vector calculus. Vector valued functions, limits and
continuity. Regular curves and differential vector calculus. Length of a curve arc. arc parameter or
arc length. Elements of differential geometry of curves (outline): tangent, normal, curvature,
twist, intrinsic triad (10 hours).
Differential calculus for real functions of several variables. Graphics and level sets. Limits and continuous
functions of several variables. Topology in Rn and properties of continuous functions (10 hours).
Partial derivatives, tangent plane, differential, directional derivatives. Higher order derivatives and successive approximations.
Partial differential equations and classification of second-order equations (introduction) (10 hours).
Unconstrained optimization (5 hours).
Differential calculus for vector-valued functions of several variables. Vector-valued functions of several variables: generalities. Surfaces in parametric form. Limits, continuity and differentiability for functions f: Rn → Rm. Smooth surfaces in parametric form. Coordinate transformations and their inverses (10 hours).
Constrained optimization (5 hours).
Integral calculus for functions of several variables. Double integrals. The calculation of triple integrals (10 hours).
Vector fields. Field lines. Gradient, divergence and curl. Differential forms and
work. Line integrals of the second kind. Irrotational, solenoidal, conservative fields. Potential.
Gauss-Green formula in the plane. Area and surface integrals. Surface integral of a vector field (flux). Divergence (or Gauss) Theorem. Curl (or Stokes) Theorem (10 hours).
Sequences of functions. Pointwise and uniform convergence (10 hours).
Series of functions and total convergence. Power series and Taylor series. Trigonometric series and
Fourier series. Pointwise and total convergence of Fourier series (10 hours).
Prerequisites
Geometry and Mathematical Analysis 1: elements of Linear Algebra (vectors, matrixes, and operations); elements of Plane and Space Geometry (lines, planes, conics, quadrics); elements of theory of linear transformations; Elements of Mathematical Analysis (complex numbers, sequences, series, limits, derivatives, functions and graphs, Taylor polynomials, definite, indefinite and improper integrals, ordinary differential equation).
Books
M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 2. Zanichelli, 2009.
M. Amar, A.M. Bersani: ANALISI MATEMATICA I – Esercizi e richiami di teoria. Amazon 2022.
M. Amar, A.M. Bersani: ANALISI MATEMATICA II – Esercizi e richiami di teoria. Amazon 2022.
D. Andreucci, A.M. Bersani: RISOLUZIONI DI PROBLEMI D’ESAME DI ANALISI MATEMATICA II. Esculapio, 1998.
online supplementary educational materials on the web page
http://www.dmmm.uniroma1.it/~alberto.bersani/A2.htm
Teaching mode
The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.
Frequency
The attendance is optional, but strongly recommended
Exam mode
At the end of the course, starting from June, the written exam (lasting 2 hours and half) consists of 5 exercises concerning the topics of the programme, with the aim of testing the knowledge level of the several techniques useful to solve the problems of the final test.
If the written exam receives an evaluation greater or equal to 15/30, the theory exam (written - 1 hour and half - or oral - 1 hour -, depending on student's request), consists of 2 theory questions (definitions, theorem statements, proofs, examples, counterexamples), starting from the results and mistakes in the written exam, in order to test the knowledge level of the theoretical aspects of the course.
The final score is a weighted mean between the scores of the written exam and of the oral exam. In both the tests the student must obtain a score greater or equal to 15/30.
Bibliography
M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 2. Zanichelli, 2009.
M. Amar, A.M. Bersani: ANALISI MATEMATICA I – Esercizi e richiami di teoria. Amazon 2022.
M. Amar, A.M. Bersani: ANALISI MATEMATICA II – Esercizi e richiami di teoria. Amazon 2022.
D. Andreucci, A.M. Bersani: RISOLUZIONI DI PROBLEMI D’ESAME DI ANALISI MATEMATICA II. Esculapio, 1998.
online supplementary educational materials on the web page
http://www.dmmm.uniroma1.it/~alberto.bersani/A2.htm
Lesson mode
The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.
Bruno Antonio Cifra
Lecturers' profile
Program - Frequency - Exams
Course program
Sequences and series of functions
Fouries series
Multiple integration and applications
Prerequisites
Numerical sequences and series
one-dimensional Integral calculus
Improper integrals
Books
Bramanti-Pagani-Salsa Analisi Matematica 2 Ed. Zanichelli
Frequency
The attendance is optional, but strongly recommended
Exam mode
Written and oral exam
Bruno Antonio Cifra
Lecturers' profile
Program - Frequency - Exams
Course program
Sequences and series of functions
Fouries series
Multiple integration and applications
Prerequisites
Numerical sequences and series
one-dimensional Integral calculus
Improper integrals
Books
Bramanti-Pagani-Salsa Analisi Matematica 2 Ed. Zanichelli
Frequency
The attendance is optional, but strongly recommended
Exam mode
Written and oral exam
- Lesson code1015376
- Academic year2024/2025
- CourseInformation Engineering
- CurriculumGestionale
- Year1st year
- Semester2nd semester
- SSDMAT/05
- CFU9
- Subject areaMatematica, informatica e statistica