27993 - Mathematical Analysis T-2

Course contents

THE EUCLIDEAN SPACE R^n. The vector space structure, the dot product and the euclidean norm. Open, closed, bounded, compact, connected subsets of R^n.
LIMITS, CONTINUITY AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES. Generalities on real and vector functions of several real variables. Definition of a continuous function and of  limit of a function. The Weierstrass theorem and the intermediate value theorem for functions of several variables. Partial and directional derivatives. Differentiable and C^1 functions; the differential and the Jacobian matrix. The chain rule. Partial derivatives of higher order. Taylor's formula of the second order for functions of several variables. Interior and constrained local extrema for real functions of several variables.
MULTIPLE INTEGRALS. Definition of Riemann double integral for functions defined on a compact rectangle. Properties of the double integral. Generalizations to more general domains. Double integrals on rectangular and normal domains computed by iterated integrals. The change of variables theorem for a double integral. Generalizations to triple integrals. Outline of double improper integrals.
CURVE AND SURFACE INTEGRALS. Smooth and piecewise smooth curves, length of a curve, integral of a function over a curve.  The integral of a vector field over an oriented curve. Conservative vector fields and their potentials. The Green-Gauss theorem. Smooth and piecewise smooth surfaces in R^3, area of a surface, integral of a function over a surface. The flux of a vector field through an oriented surface. The divergence theorem and the Stokes theorem. 
SERIES OF FUNCTIONS AND INTEGRALS DEPENDING ON A PARAMETER. Generalities on series of functions.  Power series in R and in C: Abel lemma, radius of convergence, propeties of the sum of a power series, real and complex analytic functions. Fourier series in real and complex form, propeties of Fourier coefficients, Bessel inequality. Criteria of pointwise convergence of Fourier series. Outline on mean convergence and Parseval equality. Integrals (also improper) depending on a parameter. Continuity and differentiability of integrals depending on a parameter. Examples of integrals depending on a parameter: the integral transforms.
DIFFERENTIAL EQUATIONS. The Cauchy problem for differential equations and systems. Theorems on existence, uniqueness,  continuation and continuous dependence from the data of solutions. Linear differential equations solvable by series.  Some boundary value problems for linear second order differential equations.

Teaching methods

Lessons and exercises in classroom.

Assessment methods

Written and oral examination. The written examination may be replaced by two partial written examinations.

Links to further information

http://www.dm.unibo.it/~obrecht/

Office hours

See the website of Enrico Obrecht