The course "Mathematical Analysis II" aims to provide students with some basic mathematical tools useful for engineering and application-oriented topics.
Topics of this course include parametric curves, functions of several variables and limits, partial derivatives, differentiability, gradient vector, Hessian matrix, unconstrained optimization in several variables, first order differential equations, higher order linear differential equations, multiple integration with change of variables across different coordinate systems, line and surface integrals, series of real numbers, series of functions.
Lectures for both theory and practices (exercises).
1. Parametric curves.
2. Functions of several variables (scalar and vectorial fields), limits and continuity. Directional derivatives. Differentiable functions. Necessary Necessary and sufficient conditions for differentiability. Derivatives of composite functions. Derivatives of higher order, Schwarz Theorem and Taylor polynomial in several variables. Unconstrained maxima and minima of scalar fields, necessary and sufficient conditions, Hessian matrix.
3. Metric spaces and sequences of functions (hints). Fixed-point Theorem (hints)..
4. Differential equations of the first order, with separable variables, linear, homogeneous, Bernoulli and Riccati types. Existence and uniqueness theorem (hints). Linear differential equations. Linear differential equations of higher order with constant coefficients, homogeneous and non-homogeneous,
5. Integration of functions of several variables. measurable sets. Double integrals. Surface areas and integrals. Triple integrals. Applications, center of gravity.
6. Numerical series. Convergence criteria for constant sign numerical series. Numerical alternating series and absolutely convergent series. Series of functions. Pointwise, absolute, uniform and total convergence criteria. Derivation and integration of several functions (hints). Power series, radius of convergence. Taylor series.
Handouts: "MATHEMATICS II" by prof. Maurizio Romeo, downloadable for free from the web page of the course. Sheets containing links to web pages with different solved exercises, downloadable for free from the web page of the course. Workbook: Laura Recine - Maurizio Romeo, Esercizi di analisi matematica - Volume II, Maggioli Editore.
Ricevimento: Wednesday, h. 15-16, or by appointment.
ENRICO CALCAGNO (President)
ERNESTO DE VITO (President)
CLAUDIO ESTATICO (President)
MAURIZIO ROMEO
MATHEMATICAL ANALYSIS II
Written exam and oral exam.
The written exam consists in the resolution of four exercises regarding four different topics of the course. The oral exam focuses mainly on the theory (i.e., definitions, theorems and proofs), but in some cases you will be asked to solve an exercise of a type already studied and solved during the lessons. The oral examination is restricted to students who have previously passed the written test.
Prerequisitesi: Analisi Matematica I.
Attendance: Recommended.
