UNIVERSITA’ DEGLI STUDI MEDITERRANEA DI REGGIO CALABRIA

Subject Code Subject Name Professor

56T048 Mathematical Analysis II Roberto Livrea

Department: Degree course: Class: Type of educational activity: Disciplinary Area: Scientific-Disciplinary Sector:

DICEAM Civil and Environmental Engineering L-7 Basic Mathematical Analysis MAT/05 Mathematical Analysis

Compulsory preliminary exams: Course Year: Semester:

Mathematical Analysis I I II

ECTS: Hours:

6 48

Synthetic description: This course provides a rigorous treatment of the fundamental concepts of differential and integral calculus for functions of a several variables, as well as aims to introduce students to a correct writing and communication of mathematics. Topics include: real numbers, limits, sequences and series of real numbers, continuity, compactness. The fundamental theorems on continuous functions, differentiation, and the mean value theorem. Riemman Integral Acquisition of knowledge on: The aim of the course is to introduce the Students to the main topics of the Mathematical Analysis for functions of a single real variable as: limits, derivatives and integrals. The Students will be supported in order to reach a suitable ability in: - the understanding of the theoretical aspects - the use of the elementary tools of the calculus. After successfully completing this course the Students should be able to communicate in an appropriate scientific language the notions learned, as well as to make a critical synthesis of them, when they are required to solve problems and exercise of different type. Evaluation method: A written test and an oral Student’s independent work During the period of the lessons will be proposed to the students questionnaires and themes for further exploration of the topics covered in the class. In this way, we intend to drive student’s independent work in order to facilitate learning, consolidating and deepening of the topics studied. In addition, this activity helps the student to successfully completing the course.

Detailed course program I. Limits and continuity of function of several variables taking values in R or R^n. Directional derivatives. Differentiable functions. Tangent space. Differentiability and continuity. Gradient formula. The Jacobian Matrix. The chain rule. Total derivative. Differentiability of C^1 functions. Implicit function theorem in two or more variables and with one or more constraints. Higher order derivatives. Schwartz theorem. Fermat theorem. Taylor’s formula. Max and min, necessary and sufficient conditions. Lagrange method and multipliers. II. Sequences of functions. Different types of convergence. Theorems of continuity, derivability. Limit under the integral sign. Series of functions. Different types of convergence. Integrate and derive for series. Taylor’s series. Fourier’s series. III. Differential equations. Introduction to the Cauchy Problem and to some types of boundary value problems. Existence and uniqueness for the Cauchy problem. Continuous dependence from the initial data. Generalities of the linear differential equations. Resolution of second order linear differential equations. IV. Multiple integrals. Reduction formula, changing variables and integration in R^2 and R^3. Polar, cylindrical and spherical coordinates. Volume of a solid of revolution. How to compute the barycenter and the moment of inertia. V. Curves and curve lenght. Tangent vector. The intrinsic reference frame. Curvilinear integrals of the first kind. Differential forms. Vector fields. Potential and conservative fields. Integrals of the second kind. Characterization of an integrable differential form. The work of a conservative field. VI. Regular surface: Tangent plane and normal vector. Area of a surface. Gauss-Green formula. Divergence theorem and Stokes’ formula. Integration by parts. Resources and main references M. Bertsch, R. Dal Passo, L. Giacomelli, Analisi Matematica, McGraw-Hill, Milano 2007. M. Bramanti C. D. Pagani S. Salsa, Analisi Matematica I e II, Zanichelli, 2009 Bologna N. Fusco, P. Marcellini, C. Sbordone, Elementi di Analisi Matematica due, Liguori Editore, Napoli 2001. Claudio Canuto, Anita Tabacco, Mathematical Analysis II, Springer 2008. Vladimir A. Zorich, Mathematical Analysis II, Springer 2008.

Further insights C. D. Pagani S. Salsa, Analisi Matematica, vol. I e II Masson, 1993 Milano. N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori Editore, Napoli 1996.

Subject Code Subject Name Professor

56T048 Mathematical Analysis II Roberto Livrea

Department: Degree course: Class: Type of educational activity: Disciplinary Area: Scientific-Disciplinary Sector:

DICEAM Civil and Environmental Engineering L-7 Basic Mathematical Analysis MAT/05 Mathematical Analysis

Compulsory preliminary exams: Course Year: Semester:

Mathematical Analysis I I II

ECTS: Hours:

6 48

Synthetic description: This course provides a rigorous treatment of the fundamental concepts of differential and integral calculus for functions of a several variables, as well as aims to introduce students to a correct writing and communication of mathematics. Topics include: real numbers, limits, sequences and series of real numbers, continuity, compactness. The fundamental theorems on continuous functions, differentiation, and the mean value theorem. Riemman Integral Acquisition of knowledge on: The aim of the course is to introduce the Students to the main topics of the Mathematical Analysis for functions of a single real variable as: limits, derivatives and integrals. The Students will be supported in order to reach a suitable ability in: - the understanding of the theoretical aspects - the use of the elementary tools of the calculus. After successfully completing this course the Students should be able to communicate in an appropriate scientific language the notions learned, as well as to make a critical synthesis of them, when they are required to solve problems and exercise of different type. Evaluation method: A written test and an oral Student’s independent work During the period of the lessons will be proposed to the students questionnaires and themes for further exploration of the topics covered in the class. In this way, we intend to drive student’s independent work in order to facilitate learning, consolidating and deepening of the topics studied. In addition, this activity helps the student to successfully completing the course.

Detailed course program I. Limits and continuity of function of several variables taking values in R or R^n. Directional derivatives. Differentiable functions. Tangent space. Differentiability and continuity. Gradient formula. The Jacobian Matrix. The chain rule. Total derivative. Differentiability of C^1 functions. Implicit function theorem in two or more variables and with one or more constraints. Higher order derivatives. Schwartz theorem. Fermat theorem. Taylor’s formula. Max and min, necessary and sufficient conditions. Lagrange method and multipliers. II. Sequences of functions. Different types of convergence. Theorems of continuity, derivability. Limit under the integral sign. Series of functions. Different types of convergence. Integrate and derive for series. Taylor’s series. Fourier’s series. III. Differential equations. Introduction to the Cauchy Problem and to some types of boundary value problems. Existence and uniqueness for the Cauchy problem. Continuous dependence from the initial data. Generalities of the linear differential equations. Resolution of second order linear differential equations. IV. Multiple integrals. Reduction formula, changing variables and integration in R^2 and R^3. Polar, cylindrical and spherical coordinates. Volume of a solid of revolution. How to compute the barycenter and the moment of inertia. V. Curves and curve lenght. Tangent vector. The intrinsic reference frame. Curvilinear integrals of the first kind. Differential forms. Vector fields. Potential and conservative fields. Integrals of the second kind. Characterization of an integrable differential form. The work of a conservative field. VI. Regular surface: Tangent plane and normal vector. Area of a surface. Gauss-Green formula. Divergence theorem and Stokes’ formula. Integration by parts. Resources and main references M. Bertsch, R. Dal Passo, L. Giacomelli, Analisi Matematica, McGraw-Hill, Milano 2007. M. Bramanti C. D. Pagani S. Salsa, Analisi Matematica I e II, Zanichelli, 2009 Bologna N. Fusco, P. Marcellini, C. Sbordone, Elementi di Analisi Matematica due, Liguori Editore, Napoli 2001. Claudio Canuto, Anita Tabacco, Mathematical Analysis II, Springer 2008. Vladimir A. Zorich, Mathematical Analysis II, Springer 2008.

Further insights C. D. Pagani S. Salsa, Analisi Matematica, vol. I e II Masson, 1993 Milano. N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori Editore, Napoli 1996.