Academic Year 2019/2020 - 2° Year
Teaching Staff: Giovanni NASTASI
Credit Value: 6
Scientific field: MAT/07 - Mathematical physics
Taught classes: 45 hours
Exercise: 15 hours
Term / Semester:

Learning Objectives

The aim of teaching is to provide theoretical skills and solving techniques for the study of sequences and series of functions, in the framework of differential calculus, for the integration of real functions of two or more real variables and ordinary differential equations.

Course Structure

Frontal teaching with classroom exercises.

Detailed Course Content

  1. Sequences of functions: generality; convergence simple and uniform; theorem on the continuity of the limit; theorems of passage to the limit under the sign of integral and derivative; Exercises.
  2. Series of functions: generality; simple, uniform, absolute and total convergence; theorem of continuity of the sum; theorems of integration and derivation by series; Exercises.
  3. Power series: generality; convergence radius; convergence theorems; criteria of convergence; Exercises.
  4. Taylor and MacLaurin series: sufficient conditions for Taylor series; Taylor expansion of the functions e^x, sen(x), cos(x), log(1-x), arctan(x), (1-x)^r and arcsen(x); Exercises.
  5. Fourier series: trigonometric series; expansion of periodic functions; convergence theorem; Exercises.
  6. Topology of R^2 basics: metric spaces; open and closed sets; internal, external and border points; accumulation points and isolated points; closure, derivative and frontier of a set; domains; connected, and compact sets.
  7. Functions of two variables: generality; limits; continuous functions; theorems of Weierstrass, Cantor and the existence of intermediate values; partial derivatives; Schwarz theorem; differentiability; derivative of a function composition; higher-order derivatives; directional derivatives; geometric interpretation of the gradient; null gradient functions on a connected set; Taylor's formula in the second order; relative maximum and minimum; Exercises.
  8. Functions of n variables: generality; limits; continuity; differentiability; partial derivatives; directional derivatives; derived rule of function compositions; relative extrema.
  9. Implicit functions: generality; Dini's theorem; implicit functions in the case of n variables and systems; constrained maximum and minimum; Lagrange multiplier theorem; Exercises.
  10. Differential equations: generality; differential equations of the 1st linear order; differential equations of the 2nd linear order; differential equations with constant coefficients; local and global existence and uniqueness to the Cauchy problem; solver methods for differential equations to separable and Bernoulli variables; Exercises.
  11. Line integrals and differential forms: regular curves; parametric representation; line integrals; characterization theorems of exact and closed differential forms; conservative vector fields; Exercises.
  12. Double and triple integrals: generality; reduction formulas; Guldino's theorem; coordinate changes for the calculation of double and triple integrals; Gauss-Green formulas; divergence theorems and Stokes's in R^2; Exercises.

Regular surfaces: definitions; tangent plane and normal direction; area of a surface; surface integrals; flow of a vector field; Gauss theorem; surfaces with Stokes's edge and theorem; Exercises.

Textbook Information

  • N. Fusco, P. Marcellini e Carlo Sbordone, Elementi di Analisi Matematica due, Liguori Editore – Versione semplificata per i nuovi corsi di laurea
  • P. Marcellini, C. Sbordone, Esercizi di Matematica, vol. 2 tomi 1, 2,3, e 4, Liguori Editore.
  • S.Salsa e A. Squellati, Esercizi di analisi matematica 2, Zanichelli