Academic Year 2020/2021 - 2° Year - Ingegneria Industriale Curriculum

MAT/05 - 9 CFU - 1° Semester

**At the end of the course, students will be able to apply the main ideas and topics of the differential and integral calculus of several variables to chemistry, physics, mechanics, materials science, and, moreover, to computer science, electronics circuits and energetic processes.**

**The course covers of a first development of the elementary notions of calculus of one variable, in the setting of the n-dimensional Euclidean spaces. More precisely, the course will cover most of the following main topics: limits and continuity of functions between n-dimensional Euclidean spaces, differential calculus an applications, approximation of functions by Taylor series and applications, Lebesgue measure and integral on domains, curves and surfaces and, finally, basic notions on Ordinary Differential Equations and solution techniques for some kind of first and second order ODE.**

**MAIN TOPICS**

*LIMITS AND CONTINUITY.*Elements of topology on normed spaces, with particular reference to the Euclidean ones; completeness and compactness and applications to minimum, maxium and intermediate value theorems.*FUNCTIONS SEQUENCE AND SERIES.*Pointwise and uniform convergence; continuity, derivability and integrability theorems, , convergenza puntuale ed uniforme and applications to functions series; pointwise, uniform and normal convergence series: power series, radius and interval of convergence, Cauchy- Hadamard's and D’Alembert's theorems; Abel's theorems; derivation and integration of power series; Taylor series, definitions e caractherization of convergence and analytic functions; examples:*e*. Elements of Fourier series.^{x}, sin x, cos x, sinh x, cosh x, log (1-x), log (1+x), arctan x, (1+x)^{a}, arcsin x*DIFFERENTIAL CALCULUS AND APPLICATIONS.*Differential calculus: directional and partial derivatives; differentiability; relationships with continuity; sufficients condtions for differentiability; differential of composite functions; higher order derivatives, Schwarz lemma; some theorem of calculus; applications of differential calculus to free optimizatio. Implicit function theorems and application to constrained optimization: Lagrange multipliers. Differential calculus on regular menifolds: curves and surfaces, boundary of a manifold, tangent space and tangent vectors, curvature. Differential forms: integration of functions over manifolds and integration of differential forms; exact and closed differential forms, and characterization; starshaped and simply connected sets.*INTEGRAL CALCULUS AND APPLICATIONS.*Integral calculus: multiple integrals and measure of sets. Interchange of limite and integrals, integrations with parameter. Iterated integrals, change of variables in multiple integrals. Integrals calculus on manifolds: measure of a manifolds, area di una varietà; integrals over curves and surfaces; Gauss-Green, divergence and Stokes theorems, and applications.*ELEMENTS OF THEORY OF DIFFERENTIAL EQUATIONS:*Differential equations, Cauchy problems and integral formulation; existence and uniqueness local and global theorems; methods to solve ordinary differential equations; separable equtions, linear differential equations, Bernoulli equation, Euler equation, second order constant coefficient equation.

**Please note that should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.**

**REFERENCES**

**THEORY:**

**- Teacher's lecture notes**

**- Supplementary text for more insights on argument not necessarily in program: M.Bramanti, "Metodi di Analisi Matematica per l'Ingegneria", Società Editrice Esculapio, Bologna, Luglio 2017 **

**EXERCISE BOOK:**

**- M.Bramanti, “Esercitazioni Analisi Matematica 2”, Progetto Leonardo - Edizioni Esculapio, Bologna, 2012**