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Mathematical Analysis
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
The module of Mathematical Analysis 2 aims to provide the foundations of differential and integral calculus for functions of two or more variables, vector calculus, differential equations, and series of functions. By the end of the course, the student should have assimilated the demonstrative and problem-solving process, acquiring the following operational skills: understanding of the theoretical concepts and ability to solve exercises on the topics covered in the course.
Course Prerequisites
In-depth knowledge of the topics covered in the course: Analisi Matematica I.
Teaching Methods
Lectures and exercises (also with the use of Matlab).
Assessment Methods
The exam consists of a single written test lasting approximately 4 hours, aimed at verifying the acquisition of the required skills and the ability to apply them to the topics covered in the courses of Mathematical Analysis 2 and Probability Theory.
The test mainly involves solving exercises; at the instructor’s discretion, theoretical questions may also be included.
To pass the exam with the minimum grade of 18/30, the student must demonstrate at least an elementary level of knowledge and skills.
A grade between 20/30 and 24/30 is awarded when the student is able to solve part of the exercises almost correctly, while still showing basic competence.
A grade between 25/30 and 30/30 (with possible honors) is awarded when the student is able to solve all exercises correctly and demonstrates good argumentative skills.
Texts
V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, G. Verzini: Analisi Matematica. Con elementi di geometria e calcolo vettoriale. Volume 2 (Apogeo)
Contents
The n-dimensional Euclidean space
n-tuples and vectors: the n-dimensional vector space. Cartesian coordinates. Unit vectors. Norm of a vector. Dot product. Cross product. Review of analytic geometry in the plane and in space. Equation of a line in the plane and in space. Equation of a plane in space. Quadric surfaces.
Functions of several variables
Real functions of several variables. Functions of two variables. Domain and codomain. Topology of R^2: interior, exterior, boundary points; open and closed sets; neighborhoods; bounded sets. Graphical representation of a function of two variables: graphs and level curves. Limits and continuity.
Differential calculus for functions of several variables
Partial derivatives and their geometric meaning. Differentiability. Tangent plane. The total differential theorem. Linearization of a function. Derivative of a composite function (chain rule). Gradient. Directional derivatives. Higher-order partial derivatives. Schwarz’s theorem. Hessian matrix. Taylor expansions (first and second order).
Local maxima and minima. Critical points. Fermat’s theorem. Saddle points. Classification of critical points: Hessian test.
Absolute maxima and minima on closed regions. Weierstrass theorem.
Multiple integrals
Double integrals over rectangular regions. Normal domains. Integrals over normal domains. Double integrals over general regions. Properties of double integrals. Computation of volumes, areas, and the mean value of a function of two variables. Change of variables in double integrals. Double integrals in polar coordinates.
Triple integrals over parallelepipeds. Triple integrals over normal domains. Computation of volumes of solids and the mean value of a function of three variables. Change of variables in triple integrals. Triple integrals in cylindrical coordinates. Triple integrals in spherical coordinates.
Integration over curves and surfaces
Curves in R^2 and R^3. Regular, simple, closed curves. Tangent vector. Length of a curve. Line integrals of the first kind.
Vector fields. Line integrals of the second kind. Work of a force. Conservative fields. Necessary and sufficient conditions for a field to be conservative. Potential function. Curl. Irrotational fields. Divergence. Green’s theorem in the plane in the circulation–curl form. Flux of a field along a curve. Green’s theorem in the plane in the flux–divergence form.
Surfaces and their parametrizations. Regular surfaces. Area of a regular surface. Surface integrals of functions of three variables. Oriented surfaces. Surface integrals of vector fields: flux of a field through an oriented surface.
Stokes’ theorem. Divergence theorem.
Ordinary differential equations
First-order ordinary differential equations. Cauchy problem. Separable-variable ODEs. First-order linear ODEs. ODEs reducible to linear or separable equations.
Second-order linear ODEs. Homogeneous second-order linear ODEs. Superposition principle. Homogeneous second-order linear ODEs with constant coefficients.
Initial value problems and boundary value problems. General solutions of non-homogeneous linear ODEs with constant coefficients. Method of undetermined coefficients. Method of variation of parameters.
Sequences and series of functions
Sequences of functions. Series of functions. Pointwise, uniform, and absolute convergence. Power series. Radius and interval of convergence. Convergence criteria. Differentiation and integration of power series. Taylor and Maclaurin series. Fourier series.
EXPECTED LEARNING OUTCOMES
With reference to the Dublin Descriptors, the expected learning outcomes are as follows:
Knowledge and Understanding
The student acquires fundamental concepts of multivariable calculus, including differential calculus, multiple integrals, and vector calculus (fields, curl, divergence, Green’s, Stokes’, and divergence theorems). The student also learns about first and second-order ordinary differential equations (ODEs) and series of functions.
Applying Knowledge and Understanding
The student is able to calculate partial derivatives, find local and constrained extrema, and compute multiple integrals in various coordinate systems. They can evaluate line and surface integrals, solve ordinary differential equations using the main analytical methods, and determine the convergence behavior of series of functions.
Making Judgments
The student can select the most appropriate resolution method, verify the validity of the hypotheses for the theorems applied, and provide a geometric or physical interpretation of the results obtained.
Communication Skills
The student is able to present the solution to a problem in a logically organized manner, using correct mathematical notation and providing sound reasoning for each logical step.
More information
All teaching materials (handouts, lecture notes, and supplementary resources) will be made available on the dedicated Microsoft Teams channel; the access code will be provided at the beginning of the course.