Analytical Methods in Engineering II

1. Sequences and series of functions.

Sequences of functions: general information; punctual and uniform convergence; continuity theorem; passage to the limit under integration and derivative. Series of functions: general information; simple, uniform absolute and total convergence; sum continuity theorem; integration and differentiation theorems by series.. Power series: general information; radius of convergence; convergence theorems; ratio and root criteria. Taylor and MacLaurin series: developability conditions; notable developments of the functions ex, sin(x), cos(x), log(1+x), arctan(x), (1+x)' and arcsen(x). Fourier series: trigonometric series; developability of periodic functions; convergence theorem.

2. Functions of 2 or more variables

Topology elements of R^2: metric spaces; open and closed sets; internal, external and border points; accumulation points and isolated points; closure, derivative and boundary of a set; domains; bounded, connected and compact sets; Functions of two variables: generalities; limits; continuous functions; Weierstrass, Cantor and existence of intermediate values ​​theorems; partial derivatives; Schwarz theorem; differentiability; derivative of a composite function; higher order derivatives; directional derivatives; geometric meaning of the gradient; functions with zero gradient on a connected; second order Taylor formula; relative maximums and minimums; exercises. Functions of n variables: general information; limits; continuity; differentiability; partial derivatives; directional derivatives; derivation rule for composite functions; relative extremes.

3. Implicit functions and bound extrema.

Implicit functions: generalities; Dini's theorem; implicit functions in the case of n variables and systems; local and global invertibility; exercises; Constrained extremes: general information; definition of bound maximum and minimum; Lagrange multiplier theorem.

4. Ordinary differential equations:

generality; Caucy problem; global and local existence and uniqueness theorems for a Cauchy problem; linear 1st order differential equations; nonlinear 1st order differential equations; solution methods for differential equations with separable variables and Bernoulli; linear differential equations of order n; method of variation of constants; linear differential equations with constant coefficients; similarity method.

5. Curvilinear integrals and differential forms:

regular curves; parametric representation; curvilinear integrals; theorems for the characterization of exact and closed differential forms; conservative vector fields.

6. Double and triple integrals:

generality; reduction formulas; Guldino theorem; coordinate changes for calculating double and triple integrals; Gauss-Green formulas; divergence and Stokes theorems in R^2.

7. Regular surfaces:

definitions; tangent plane and normal unit; area of ​​a surface; surface integrals; flow of a vector field; Gauss theorem; surfaces with boundary and Stokes theorem.

[1] N. Fusco, P. Marcellini, C. Sbordone, Elementi di Analisi Matematica 2 - versione semplificata per i nuovi corsi di Laurea, Liguori Editore (2001)

[2] M. Bramanti, C. Pagani, S. Salsa, Analisi Matematica 2, Zanichelli (2009)

[3] P. Marcellini, C. Sbordone, Esercizi di Analisi Matematica 2, Zanichelli (2017) 

[4] S. Salsa, A. Squellati, Esercizi di Analisi Matematica, Zanichelli (2011)

[5] N. Fusco, P. Marcellini, C. Sbordone, Lezioni di Analisi Matematica due, Zanichelli (2020)

Study the punctual or uniform convergence of a given sequence of functions

Write the Fourier series expansion of a periodic function

Determine the maximum and minimum points of a given function

Dini's theorem for implicit functions

Solving ordinary linear differential equations

Calculation of double integrals

Integrals of differential forms along curves

Theorem on differential forms

Continuity theorem of the sum function

Weierstrass test (Total convergence implies absolute and uniform convergence)

Radius theorem (Absolute and total convergence of power series)

Sufficient condition for Taylor series developability

Existence theorem of zeros

Weierstrass theorem

Existence of directional derivatives of differentiable functions

First order Taylor formula

Theorem on functions with zero gradient.

Euler's identity

Fermat's theorem

Uniqueness of the solution to the Cauchy problem.

The exact differential forms of class C^1 are closed.