Academic Year 2020/2021 - 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 in the field of calculus with the aim to apply them in physical or engineering courses. In particular, the target will be to develop skills about power and Fourier expansion of functions, differential calculus and integration of real functions of two or more real variables, solutions of ordinary differential equations and integration on curves or surfaces.

Course Structure

Frontal teaching with classroom exercises.

Detailed Course Content

  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 ex, 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.

Textbook Information

[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.