27213 - Mathematical Analysis 2

Learning outcomes

Multivariable calculus: theory and techniques.

Course contents

Preliminaries: R^n as a normed linear space. Closed, open, compact and bounded sets. Functions of several variables: limits, boundedness, continuity and the main properties of continuous functions.

Differential calculus: partial derivatives, directional derivatives, C^1 functions and their differentiability. Plane tangent to the graph of a function, differential, Jacobian and gradient. Chain rule and other rules of differentiation. Higher order derivatives and Hessian matrix: Schwarz Theorem. Taylor formula of 1st and 2nd degree. Local minima and maxima in multivariable calculus, necessary and sufficient conditions.

Integration Definition of measure according to Peano in R^n, exetrior and interior measure, Riemann integration. Properties of integrals: mean value theorem, Cavalieri principle, change of variables in multiple integrals.

Integration by parts in several variable calculus: surfaces, normal vector, vector fields, integration on a surface. Work and line integrals. Differential forms: exact and closed; theorem of Volterra-Poincarè. Gauss Theorem.

Linear ordinary differential equations homogeneous and non homogeneous case. The general theory and explicit solution on particular cases.

Readings/Bibliography

Marco Bramanti, Carlo Domenico Pagani e Sandro Salsa: Matematica, Zanichelli

Teaching methods

Lectures and exercise sessions.

Assessment methods

Written exam and oral exam, to be passed both within the same exam session.

Teaching tools

Lecture notes, exercises, old exams with solutions worked out in detail at

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

and links to other resources on the web.

Links to further information

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

Office hours

See the website of Nicola Arcozzi