28616 - Mathematical Analysis T-B

Learning outcomes

Knowing the methodological-operational aspects of mathematical analysis, with particular attention to the functions of multiple real variables and differential equations, in order to be able to use this knowledge to interpret and describe engineering problems.

Course contents

Real numerical series.

Basics on numerical sequences. Definition of numerical series and necessary conditions for their convergence. Criteria for convergence of series of non-negative real numbers. Leibniz Theorem.

 

Improper integrals.

Definition of improper integrals and criteria for their convergence.

 

Complex numbers.

Algebraic  and trigonometric form. Exponential form of a complex number. De Moivre's formula. Complex roots of a complex number. Algebraic equations in C.

 

The n-dimensional Euclidean space.

The structure of vector space, scalar product and Euclidean norm. Elements of topology.

 

Limits, continuity and differential calculus for functions of several real variables.

Real and vector functions of several real variables: generalities. Definition of limit and continuous function. Weierstrass theorems, intermediate values for functions of several variables. Definition of partial derivative and directional derivative. Differentiable functions and functions of class C^1; the differential and the Jacobian matrix. The theorem on the differentiability of a compound function. Higher-order partial derivatives. Second-order Taylor formula for functions of several variables. Relative extrema for free real functions of several real variables.

 

Multiple integrals.

Definition of Riemann double integral on finite and measurable sets. Properties of the double integral. Reduction theorems on rectangles and on simple sets. The theorem of change of variables. Triple integrals: extension of definitions and theorems on double integrals.

 

Ordinary Differential Equations.

Linear differential equations of first order: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Linear differential equations of second order with constant coefficients: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Generalization to variable coefficients and arbitrary order equations. Separable ODEs.

Readings/Bibliography

THEORY

1. M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica (seconda edizione), McGraw-Hill (2011).
2. G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, Zanichelli (2009).
3. G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, Zanichelli (2015).

EXERCISES

1. M. Bramanti: Esercitazioni di Analisi Matematica 1, Progetto Leonardo - Esculapio (2011).
2. M. Bramanti: Esercitazioni di Analisi Matematica 2, Progetto Leonardo - Esculapio (2012).

Teaching methods

Lectures and weekly student reception.

Assessment methods

At the end of the course there will be a written and an oral test, both of which are compulsory.

The written test will consist of exercises. It will last 2 hours and 30 minutes. It is forbidden to use notes nor books.

Access to the subsequent oral test is allowed only to those who have passed the written test with a score greater than or equal to 18/30.

The oral test will focus on verifying understanding of the fundamental concepts and knowledge of the definitions and statements of the main results, examples and counterexamples.

There will be 6 rounds (each of which will have both written and oral test): 4 in the summer session (3 between June-July + 1 in September) and 2 in the winter session (January-February).

The written and the oral test must be taken within the same round.

Teaching tools

Sheets of exercises will be made available, uploaded on the UniBo "VIRTUAL" website.

Office hours

See the website of Eugenio Vecchi