27213 - Mathematical Analysis 2

Learning outcomes

At the end of the course the student will master the basic results as well as the basic tools of advanced multivariable calculus. He/She will master the notions of differentiability and integrability 'for functions of several real variables. He/She will be able to apply this knowledge to the solution of problems posed by the pure and applied sciences.

Course contents

FUNCTIONS OF SEVERAL VARIABLES:
Topology of R^n. Real-valued and vector-valued functions of several variables: generalities, limits and continuity. Differential calculus. Local invertibility, the Implicit function theorem. Maxima and minima, manifolds of R^n, critical points on manifolds of R^n (Lagrange multiplier theorem).

CURVES AND line INTEGRALS: Definition of a regular curves, length of a curve and Integral of a function over a curve.

MULTIPLE INTEGRALS: Integral of a continuous function. Properties of the integral. Reduction and change of variable theorems for multiple integrals.

DIFFERENTIAL FORMS AND VECTOR FIELDS: Differential forms and vector filelds. Integral of differential forms and vector fiellds. Potentials. The Gauss-Green formula.

SURFACES: Regular surfaces. Tangent plane. Density area a surface. Surface integrals. The Divergence Theorem and the Stokes' formula.

ORDINARY DIFFERENTIAL EQUATIONS: First order linear equations, linear equations with constant coefficients, equations with separable variables.

SEQUENCES AND SERIES OF FUNCTIONS: Convergence, uniform convergence, total convergence. Power series, Taylor series and Fourier series: generalities.

Readings/Bibliography

E. Giusti: Analisi Matematica 2 (Boringhieri);

C.D. Pagani, S. Salsa: Analisi Matematica 2 (Zanichelli).

Teaching methods

Lectures and exercises given by the teacher in the classroom using the blackboard.

Assessment methods

Written exam, made of exercises and theoretical questions. Oral exams only in case of the summa cum laude.

Office hours

See the website of Alberto Parmeggiani