Mechanical Engineering for the Green Transition

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).

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).

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

The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.

The attendance is optional, but strongly recommended

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.

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

The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.

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).

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).

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

The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.

The attendance is optional, but strongly recommended

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.

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

The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.

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).

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).

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

The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.

The attendance is optional, but strongly recommended

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.

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

The courses are based on theoretical lessons and exercises, given by the thacher, together with exercises classes given by a tutor.

Sequences and series of functions Fouries series Multiple integration and applications

Numerical sequences and series one-dimensional Integral calculus Improper integrals

Bramanti-Pagani-Salsa Analisi Matematica 2 Ed. Zanichelli

The attendance is optional, but strongly recommended

Written and oral exam