Learning objectives
Students must demonstrate sufficient knowledge and understanding of the basic results of multivariable calculus and ordinary differential equations. Lectures emphasize concrete computations over more theoretical considerations with little emphasis on exceptional cases.
In particular, students must
1. exhibit sufficient conceptual understanding and good computational fluency in standard cases;
2. be able to exploit tools from multivariable calculus and ordinary differential equations for solving problems ranging from simple to medium difficulty;
3. be able to evaluate coherence and correctness of results obtained by themselves or by others;
4. be able to communicate in a clear and precise way the topics of lectures using the appropriate scientific lexicon;
5. be able to read autonomously scientific and technical books and articles which exploit multivariable calculus and ordinary differential equations.
Prerequisites
Solid knowledge of single-variable calculus and linear algebra.
Course unit content
Multivariable calculus and ordinary differential equations.
Full programme
1) Linear algebra and topology. Linear algebra and geometry: vector spaces, norm, scalar product and Cauchy-Schwarz inequality; matrices, eigenvalues and diagonal form of symmetric matrices, quadric forms; basic results of analytical geometry in space. Topology: interior, limit and bundary points; open and closed sets; compact sets and connected sets. 2) Multivariable differential calculus. Limits and continuity: limits for functions of several variables; continuous functions of several variables; Weierstrass' and intermediate values theorems. Multivariable calculus: directional and partial derivatives, differentiability of scalar and vector valued functions, gradient; tangent plane, tangent and normal vectors; chain rule;functions of class C^1; inverse function theorem, diffeomorphisms and change of variables. Functions of class C^2: Schwarz's theorem and hessian matrix; second order Taylor's formula; local and global maxima and minima, saddle points; necessary and sufficient conditions for optimality. Surfaces: implicit function theorem, Lagrange's multipliers. 3) Curves and vector fields. Curves: simple, closed and smooth curves, length of a smooth curve. Vector fields: line integral; potentials; irrotational vector fields. 4) Multiple integrals Integration: measure of sets; definition of integral and of integrable functions; dimensional reduction and Fubini--Tonelli's theorem. Change of variable formula: geometrical meaning of jacobian, spherical and cylindrical coordinates. 5) Ordinary Differential Equations Ordinary differential equations: definitions and examples; local existence and uniqueness of solutions; maximal and global solutions; solution methods for linear, separable and Bernoulli's equations. Second order linear differential equations: fundamental system of solutions, Lagrange's variation of parameters.
Bibliography
Handouts and material taken from the following textbooks:
N. Fusco -- P. Marcellini -- C. Sbordone, "Lezioni di analisi matematica due", Zanichelli, Bologna 2020;
C.D. Pagani -- S.Salsa, "Analisi Matematica Vol. 1 e 2", Zanichelli, Bologna 2015.
Teaching methods
In-person instruction (4 hours per week) and online exercise sessions (2 hours per week). Students must comply with the university's safety guidelines to mitigate the spread of covid-19. Lectures will be recorded and will be available on the Elly-Dia web page for the entire week following that of recording.
Assessment methods and criteria
Assessments consists of a written test supplemented by an oral examination. There are no intermediate tests.
The written test consists of exercises and multiple choice questions. The oral examination is subject to successful completion of the written test (grade 16/30 or higher). It consists of reviewing the written test and interviewing the candidate on the course contents. The final grade takes into account both parts.
Other information
The course will be quite fast-paced and it is essential to work steadily throughout the semester.
