SM/0026 - MATHEMATICAL ANALYSIS 2
Academic Year 2021/2022
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FABRIZIO CUCCU (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/64] MATHEMATICS||[64/00 - Ord. 2017] PERCORSO COMUNE||10||80|
KNOWLEDGE AND UNDERSTANDING
Learning multivariable differential calculus. Limits, continuity, derivatives and differentiability. Multiple integrals, line and surface integrals.
APPLYING KNOWLEDGE AND UNDERSTANDING
Students will be able to apply the theory to solve free and constrained optimization problems in several variables (in particular for two and three variables),
to evaluate multiple, line and surface integrals, and to study differential linear differential forms in R^2 and R^3.
The course has an approach both intuitive (with applications, mainly from Physics) and rigorous (proofs) of the material covered. This is done in order to develop as much as possible students' critical thinking.
Students will be able to explain the material covered, both conceptual (statements and proofs) and computational (differential and integral calculus) topics.
Students will learn differential calculus in several variables and some related statements and proofs they need to complete their studies in Mathematics.
Knowledge of the algebraic and topological structure of real numbers, elementary linear algebra, analytic geometry in plane and space, single variable calculus (including sequences and series).
SOME NOTES ON ORDINARY DIFFERENTIAL EQUATIONS
First order linear equations; Second order linear equations with constant coefficients, the method of variation of constants; separation of variables.
BASIC TOPOLOGY IN R^n
Open balls, open sets, closed sets, accumulation points, sequences in R^n, compact sets, connected open sets.
FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES
Graph of a function, level sets (contour lines and contour surfaces), limits, continuity, partial derivatives, gradient, Hessian matrix, differentiability, differential,
derivatives of compound functions (chain rule), directional derivatives, Lagrange's mean-value theorem, Taylor's formula, second differential. Local maxima and minima, stationary points and necessary condition, second order conditions.
Component functions, limits, continuity, derivatives, Jacobian matrix, differentiability, derivatives of compound functions (chain rule) , Jacobian determinant.
CURVES AND INTEGRALS OF FUNCTIONS OVER CURVES
Definition of curve, regular curves, equivalent curves, oriented curves, length of a curve, integral of a scalar function over a curve.
LINEAR DIFFERENTIAL FORMS
Dual space of R^n, linear differential forms, work of a vector field, line integral of linear differential forms, exact forms, closed, primitive functions, conservative vector fields, irrotational vector fields, potential function, simply connected open sets, linear differential forms in a simply connected open set,
computation of primitive functions.
DOUBLE AND TRIPLE INTEGRALS
Normal sets in R^2, area of a normal set, definition of integrable function, double integrals, integrability of continuous functions, reduction formulas for double intregrals. Regular sets in R^2, positive orientation of the boundary of a regular set, Gauss-Green formulas, change of variables formula in R^2, polar coordinates. Normal sets in R^3, volume of a normal set, definition of integrable function, triple integrals, integrability of continuous functions, reduction formulas for triple intregrals, method of slicing, change of variables formula in R^3, cylindrical and spherical coordinates.
SURFACES AND SURFACE INTEGRALS
Definition of parametrized regular surface, equivalent regular surfaces, area of a regular surface, orientable surfaces and their orientation, regular surfaces with boundary, surface integral of a function, flux of a vector field across a surface, Stokes' formula, the divergence theorem.
THE IMPLICIT FUNCTION AND INVERSE FUNCTION THEOREMS
The implicit function theorem, the inverse function theorem, tangent line (respectively, plane) to an implicitly defined curve (respectively , surface).
CONSTRAINED MAXIMA AND MINIMA
Definitions of constrained local extremum, constrained global extremum, constrained stationary point and their relationship (the method of Lagrange multipliers).
The course consists of lectures (for a total amount of 80 hours) using the blackboard in which theoretical aspects, some applications and exercises are explained.
Subject to availability, a tutoring program of 40 hours is provided. Attendance at this activity is not compulsory.
Verification of learning
The assessment consists of a final exam split in two parts: a written test and an oral exam. Passing the written test is required to take the oral exam.
The written test consists of five problems and aims to test the ability in computation of partial derivatives, solving optimization problems, evaluating multiple, line and surface integrals, studying linear differential forms in plane and space. The maximum time allowed is two hour and thirty minutes. Using books or notes is not permitted (nevertheless a formulary on trigonometric and hyperbolic functions is allowed). The assessment is based both on how the student is able to manage problems and on their mastery of calculation. The grade is proportional to the amount of problems correctly solved. The test is considered passed if the grade is at least 18/30.
The oral exam is agreed upon contacting the professor personally, by telephone or by email. The oral exam consists mainly of some questions about conceptual topics of the course (definitions, theorems, proofs and important examples). The oral exam lasts about an hour and it is evaluated considering the following aspects: acquisition of the subjects mentioned in the objectives, logic and communication skills.
If the oral is passed then the final grade is approximately equal to 30% of the written test grade plus 70% of the oral exam grade, if it is failed then the written test must be resat.
N. Fusco, P. Marcellini, C. Sbordone. Analisi Matematica Due, Liguori editore
P. Marcellini, C. Sbordone. Esercitazioni di Matematica, vol.2, parte prima e seconda, Liguori editore
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