### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
GIUSEPPE VIGLIALORO (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/60]  PHYSICS [60/00 - Ord. 2012]  PERCORSO COMUNE 6 48

### Objectives

•  Knowledge and understanding: Knowledge and understanding of theoretical principles of differential and integral calculus for functions of several variables. Knowledge and understanding of the fundamentals of the theory of ordinary differential equations.

•  Applying knowledge and understanding: Ability in the usage of the differential and integral calculus of functions of several variables in order to find limits, maxima and minima of simple functions of two variables, as well as to compute circuitations and flux of a given vector field along specific curves and surfaces. Ability in the usage of the theory of ordinary differential equations in finding the general solution of a given equation and the particular solution of a simple initial-value problem.

•  Making judgements: Ability in the physical or geometrical interpretation of the data and the results even as a tool for checking their plausibility.

• Communication skills: Ability in communicating information, ideas, problems and solutions based on differential and integral calculus for functions of several variables and on the theory of ordinary differential equations.

• Learning skills: Developing learning skills necessary to understand the theoretical principles of differential and integral calculus for functions of several variables and the fundamentals of the theory of ordinary differential equations.

### Prerequisites

Analytic geometry, trigonometry, differential and integral calculus of functions of one real variable.

### Contents

• Ordinary differential equations. First order differential equation: separable equations, linear, Bernoulli equations. Cauchy theorem for the existence and uniqueness of the solution. Linear equations of order n. Wronskian determinant. Linear equations with constant coefficients: Lagrange method and special methods for determining a particular solution of the nonhomogeneous equation.

• Plane and spatial curves. Parameters, trace, regular and piecewise-regular curves, tangent vector, closed curves (cycles). Rectifiable curves, arc length, length of a regular curve. Line integrals.

• Differential calculus of functions of several variables. The topology of plane and space. Limits and continuity. Partial derivatives, gradient, differentiability, tangent plane, directional derivative. Higher-order partial derivatives. Equality of mixed partials (Clairaut's theorem, also called Schwarz's theorem). Jacobian matrix. Hessian matrix. Plane and spatial polar coordinates. Cylindrical coordinates. Second-order Taylor's expansion.

• Optimization. Free extrema of functions of several variables. Necessary condition (gradient must vanish). The role of the Hessian matrix. Constrained optimization.

• Integral calculus of functions of several variables. Double integral over regular domains: definition, geometrical interpretation, elementary properties. Computation of double integrals: reduction to two simple integrals, change of variables. Triple integrals: elementary properties and computations.

• Vector fields and line integrals. Line integral of the second kind. Gradient, divergence and curl for fields in space. Conservative fields. Potential. Gauss-Green formulae in the plane.

• Parametric surfaces. Parametric representation of a surface. Regular surfaces. Area. Surface integral. The divergence theorem. The Stokes theorem.

• Sequences and series of functions. Pointwise convergence. Uniform convergence. Power series.

### Teaching Methods

Teaching will be carried out in presence, integrated and "complemented" with online activities, in order to guarantee an innovative development.

Lectures (30 hours), exercises and discussion with students (18 hours)

### Verification of learning

The exam, held in presence, consists of a written test in which the following topics are proposed: general analysis of real functions of real variables, integral calculus and applications, differential equations and vector fields. The student will have to demonstrate that he has understood and learnt the techniques for handling and exposing each of the topics discussed and to know how to apply the various methodologies linked to the techniques of resolution. The exam score is awarded by a vote between 1 and 30. The test consists of 5 exercises, 4 of which are mandatory (with score 7.5 each) and 1 optional (with score 3) and the score is determined according to the following rule: sum of the scores obtained in the individual exercises. In evaluating the examination, the final vote determination takes into account the logic followed by the student for each proposed exercise, the calculation strategy chosen in terms of the hypothesis of the problem, the clarity of the exposition and the reasoning.

### Texts

Book:

• Analisi Matematica due. Liguori Editore (1992). Authors: N. Fusco, P. Marcellini, C. Sbordone.

Exercises book:

•  Esercitazioni di Matematica, vol. 2, parte prima e parte seconda, Liguori Editore (1989). Authors: P. Marcellini,  C. Sbordone.