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 and Bolzano theorems for functions of several variables. Partial derivative and directional derivative. Differentiable functions and functions of class C^1. Jacobian matrix. 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:

Enrico Giusti, Analisi Matematica 2, Terza edizione, Bollati Boringhieri

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

 

Teaching methods

Frontal lectures, resolution of exercises.

Assessment methods

Assessment methods

Oral examination

Teaching tools

Notes of the teachers in virtuale. Tutor.

Office hours

See the website of Annamaria Montanari

See the website of Matteo Franca