16314 - Mathematical Analysis B

Learning outcomes

The student should acquire techiques and methods of mathematical analysis in several dimensions and of the theory of ordinary differential equations, which are of common use in research and work in aerospace and mechanical engineering.

Course contents

THE EUCLIDEAN SPACE R^n.

The structure of vector space, scalar product and Euclidean norm. Open balls. Open sets, closed sets, bounded sets, compact sets, arc-connected subsets of R^n.

LIMITS, CONTINUITY AND DIFFERENTIAL CALCULUS

Real and vector functions of several real variables. Accumulation points. Limit of a function. Continuous functions. Weierstrass theorem for functions of several variables. Partial derivative and directional derivative. Differentiable functions and functions of class C^1. Chain rule. Partial derivatives of higher order. Hessian matrix. Maxima/minima of a function. Taylor's formula for functions of several variables. Lagrange mean value theorem. Fermat's theorem.

MULTIPLE INTEGRALS.

Definition of Riemann double integral for functions defined on a normal domain. Properties of the double integral. Double integrals on normal domains computed by iterated integrals. The change of variables theorem for a double integral. Generalizations to triple 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. Irrotational and conservative vector fields: evaluation of the potentials.

Poincare theorem on simply connected sets.The Green-Gauss theorem, the divergence theorem, Stokes formula.

DIFFERENTIAL EQUATIONS.

The Cauchy problem for differential equations. Theorems on existence, uniqueness and continuation of solutions. Solving methods for nonlinear differential equation with separable variables, for linear differential equations of the first order, for second order linear differential equations with constant coefficients

Readings/Bibliography

Theory:

Lecture Notes available online on Virtuale.

 

References used for the lecture notes:

 

1) E. Giusti- Analisi Matematica 2, Terza Edizione, Bollati Boringhieri editore.

2) Nicola Fusco, Paolo Marcellini, Carlo Sbordone. Elementi di Analisi Matematica due. Versione semplificata per i nuovi corsi di laurea. Liguori Editore

3) - G.C. Barozzi, G. Dore, E. Obrecht, Elementi di Analisi Matematica (Vol 2), Zanichelli, 2015.

 

 

Exercises:

 

E. Giusti, Esercizi e complementi di analisi matematica (Vol. 2), Bollati Boringhieri, 1992.

 

Paolo Marcellini, Carlo Sbordone. Esercitazioni di matematica Volume II, Parte prima e seconda. Liguori Editore

 

 

 

Teaching methods

The course of Mathematical Analysis B takes place in the second semester and represents the second part of the integrated course of Mathematical Analysis (12 credits). The course of Mathematical Analysis B is structured in classroom lectures, in which the theoretical aspects of the topics covered are first presented. In particular, after introducing the basic notions, the main theorems and results in the field of differential and integral calculus for functions of several real variables and ordinary differential equations are stated and demonstrated. Subsequently, ample space is dedicated to the resolution of exercises.

Assessment methods

The student may attend the exam if they has already obtained a positive evaluation in the written part of Analisi Matematica A of the same year.

The test is split in a theorical part and an exercise part to be given in sequence. In the exercise test the student should solve some exercises, while in the theorical part should answer to some open theorical questions in written form, possibly state some theorems and prove their statements rigorously.

 

Once the exam is positively evaluated, the student may ask for an oral integration of the exam.

To attend the exam  the student should sign in the list available in AlmaEsami [https://almaesami.unibo.it/] (at least 3 days in advance).

The student may accept a positive mark or he may repeat the exam.

The global mark is the average between the marks obtained in Analisi Matematica A e Analisi Matematica B. The mark in Analisi Matematica A is in force for 12 months and if the student has two different marks in the same modulus only the more recent mark will be considered.

 

Once the exam has been taken, the grades will be uploaded to the appropriate platforms (almaesami / virtual) and you will be given at least one week to reject the grade, after which the exam will be considered accepted and the grade received will be considered confirmed, even if there was a better previous grade.

 

 

Teaching tools

Notes of the teacher [https://virtuale.unibo.it/].

 

Office hours (Tuesday afternoon in presence or online, by appontment).

 

Travaux Dirigees directed by the tutor.

Office hours

See the website of Simone Ciani