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. 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:

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

Exercises:

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

Teaching methods

Frontal lectures, resolution of exercises.

Assessment methods

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

The test is split in an exercise part and a theorical part to be given in sequence: firstly the student should clear the exercise part with a mark of 15 or more, and then he may attend the theorical part. 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. 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 2 days in advance).

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.

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

 

Teaching tools

Notes of the teacher and exercises of the tutor and recordings of the lessons [https://virtuale.unibo.it/].

Office hours

See the website of Matteo Franca