Analytical Methods in Engineering II
Academic Year 2025/2026 - Teacher: ANDREA GIACOBBEExpected Learning Outcomes
Course Structure
Required Prerequisites
Attendance of Lessons
Detailed Course Content
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; 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.
Textbook Information
[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)
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Sequences and series of functions | |
| 2 | Functions of more than one variable | |
| 3 | Implicit functions and constrained extremes | |
| 4 | Ordinary differential equations | |
| 5 | Path integrals and differential forms | |
| 6 | multiple integrals | |
| 7 | Regular curves and Stokes theorem |
Learning Assessment
Learning Assessment Procedures
Examples of frequently asked questions and / or exercises
In a folder of a Teams class the student will find many past written exams. The oral exam will consist of a question taken among those in the following list:
1. Sequences of functions; pointwise convergence, absolute convergence, uniform convergence, relations among the above types of convergences.
2. Theorems on continuity and integrability of limits of sequences of functions.
3. Series of functions; convergence, absolute convergence, uniform convergence, total convergence, relations among the above types of convergences.
4. Power series and their convergence. Taylor series.
5. Limits for functions of two variables: definition, bridge theorem, examples.
6. Differential of functions of two variables.
7. Directional derivatives: definitions and their relations with the differential.
8. Gradient and Hessian matrix of a function, relations with the graph of the function.
9. Dini's Theorem on implicit functions.
10. Theorem of Fubini and Tonelli and its use to compute integrals in higher dimension.
11. Change of variable in multiple integrals. The polar coordinates.
12. Separable differential equations: definitions and solutions.
13. Linear differential equations of order I and II: definitions and solutions.
14. Bernoulli differential equation: definitions and solutions. Other families of differential equations that can be reduced to separable differential equations.
15. Differential forms.
16. Stokes Theorem and applications to the 2 and 3-dimensional case.
17. Baricenters, length of curves, areas enclosed in curves.