METODI ANALITICI PER L'INGEGNERIA II

Frontal teaching with classroom exercises.

1. Sequences and series of functions.
   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.
   Series of functions: generality; simple, uniform, absolute and total convergence; theorem of continuity of the sum; theorems of integration and derivation by series; Exercises.
   Power series: generality; convergence radius; convergence theorems; criteria of convergence; Exercises.
   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.
   Fourier series: trigonometric series; expansion of periodic functions; convergence theorem; Exercises.
2. Functions of two or more variables.
   Topology of R² 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.
   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.
   Functions of n variables: generality; limits; continuity; differentiability; partial derivatives; directional derivatives; derived rule of function compositions; relative extrema.
3. Implicit functions and constrained extrema.
   Implicit functions: generality; Dini's theorem; implicit functions in the case of n variables and systems; Exercises.
   Constrained extrema: constrained maximum and minimum; Lagrange multiplier theorem; Exercises.
4. Ordinary differential equations: generality; Cauchy problem; local and global existence and uniqueness to the Cauchy problem; linear differential equations of the 1st order; nonlinear differential equations of the 1st order; solver methods for differential equations to separable and Bernoulli variables; linear differential equations of order n; variation of parameters method; differential equations with constant coefficients; Exercises.
5. Line integrals and differential forms: regular curves; parametric representation; line integrals; characterization theorems of exact and closed differential forms; conservative vector fields; Exercises.
6. 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 R²; Exercises.
7. 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.

[1] N. Fusco, P. Marcellini e C. Sbordone, Elementi di Analisi Matematica due – Versione semplificata per i nuovi corsi di laurea, Liguori Editore, 2001 – ISBN: 9788820731373
[2] M. Bramanti, C. D. Pagani e S. Salsa, Analisi matematica 2, Zanichelli, 2009 – ISBN: 9788808122810
[3] P. Marcellini e C. Sbordone, Esercitazioni di Analisi matematica Due, Zanichelli, 2017 – Prima parte ISBN: 9788808220707 e Seconda parte ISBN: 9788808191458
[4] S.Salsa e A. Squellati, Esercizi di Analisi matematica, Zanichelli, 2011 – Volume 2 ISBN: 9788808218964

Other textbooks

[5] N. Fusco, P. Marcellini e C. Sbordone, Lezioni di analisi matematica due, Zanichelli, 2020 – ISBN: 9788808520203

English textbooks

[i] A. Friedman, Advanced Calculus, Dover Publications, 1999.
[ii] R. Wrede, M. R. Spiegel, Advanced Calculus Third Edition, Schaum's Outline Series, McGraw-Hill, 2010.
[iii] R. A. Adams, C. Essex, Calculus: A complete course - 7th ed., Pearson, 2010.