The space R^n.
Differential calculus for (scalar- and vector-valued) functions of several variables.
Free and constrained optimization problems.
Curves. curvilinear integrals.
Parametric surfaces, surface and flow integrals.
Function spaces. Sequences and series of functions, power series.
Suggested textbook (theoretical part)
C. Pagani, S. Salsa, Analisi matematica Vol. 2, Zanichelli 2009.
Suggested textbooks (exercises)
S. Salsa, A. Squellati, Esercizi di Analisi Matematica 2, Zanichelli, 2011.
P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica 2, prima e seconda parte, Liguori, 2017.
Other textbooks (theoretical part)
N. Fusco, P. Marcellini, C. Sbordone, Analisi matematica 2, Liguori, 2016
Knowledge: differential and integral calculus for functions of several variables; free and constrained optimisation for functions of several variable; curves and surfaces; sequences and series of functions; Fourier series.
Skills: autonomy in proposing, articulating and rigorously supporting arguments for the resolution of problems related to the listed subjects (see Knowledge); confident use of symbols and main results; control of errors.
Abilities/capacities: consolidated communication skills in writing.
Differential and integral calculus for single-variable functions. Numerical sequences and infinite series. Basics on Ordinary Differential Equations. Linear algebra and analytic geometry.
Prerequisite (formal): "Analisi Matematica I". Recommended: "Geometria".
Lessons (in the classroom), in the absence of a rigid separation between theory and practice.
Type of Assessment
The final examination is formed of two written parts. The first part consist in solving some exercises. In the second part the student is required to answer some theory questions concerning the content of the course. The second part is reserved to those students who passed the first part. The final mark is an appropriate mean of the marks of the two parts.
The n-dimensional Euclidean space. Scalar product, Euclidean norm, Cauchy-Schwartz inequality, subadditivity of the norm. Vector product. Topology: open, closed and compact sets.
Functions of several variables. Limits; continuity. Partial derivatives, gradient, directional derivatives. Differentiability. Second derivatives, Hessian matrix. Optimisation: point of relative and absolute maximum and minimum. Techniques to identify free and constrained maxima and minima.
Integral calculus for functions of several variables (only the cases of two and three variables). Definition and fundamental properties of the integral. Integrability of some classes of functions. Normal domains and reduction formulas for multiple integrals. Change of variables in multiple integrals; polar coordinates.
Parametric curves. Support and orientation. Regular curves, tangent vector. Length of a curve and curvilinear integrals. The Gauss-Green formulas.
Surfaces in the three-dimensional space. Regular surfaces; tangent plane and normal vector. Area of a surface and surface integrals. The divergence theorem.
Sequences and series of functions. Point-wise and uniform convergence. Total convergence of series of functions. Power series. Fourier series.