Teacher
|
PIERRI ANNA
(syllabus)
LEARNING OBJECTIVES. The teaching presents the basic elements of Mathematical Analysis 2. The educational objectives of the teaching involve the acquisition of demonstrative results and techniques, as well as in the ability to use the related computational tools DIDACTIC DELIVERY NO. 48 ONLINE VIDEO LECTURES (NO. 6 TEACHING UNITS - LASTING TWO HOURS PER CFU)
INTERACTIVE DIDACTICS NO. 2 INTERACTIVE LECTURES PER CFU. N. 5 THEMATIC DISCUSSIONS ON DIDACTIC FORUM (TOPIC) AND N. 2 POSTS PER CFU AS PER PQA TEACHING GUIDELINES. NO. 2 E-ACTIVITIES PER 5 CFU. NO. 2 TESTS PER CFU WITH 8 MULTIPLE CHOICE QUESTIONS.
COURSE SYLLABUS 1. Indefinite and definite integrals: Primitives and indefinite integral. Immediate integrals of elementary and compound functions. Rules and methods of integration. Integrals by parts and by substitution. Definite integral and its geometric meaning. Averaging theorem. Integral function and fundamental theorem of integral calculus. Fundamental formula of integral calculus. 2. Numerical series: Link between numerical successions and numerical series. Converging, diverging and indeterminate series. Geometric, harmonic and telescopic series. Convergence criteria for series with positive terms: comparison, ratio and root criteria. Leibniz criterion for convergence of series with alternate sign. 3. Functions of two variables: Domains and topology for functions of two variables. Limits, continuity and derivability. Differentiability. Relative maxima and minima for functions of two variables: generalities; necessary condition of first order. Sufficient condition of second order. 4. Differential equations: Cauchy problem of first and second order. Cauchy theorem for first-order differential equations. First-order differential equations: linear with variable coefficients; with separable variables; homogeneous; of the type f(ax+by). Second-order differential equations with constant coefficients. 5. Series of functions: Point, absolute, uniform and total convergence of a series of functions. Power series: generalities; radius of convergence and set of convergence; Cauchy - Hadamard theorem; D'Alembert's theorem. Fourier series: generalities and conditions of developability; calculation of Fourier coefficients for functions of any type, even functions and odd functions. Bessel inequality and Riemann - Lebesgue lemma. Pointwise convergence theorem. Uniform convergence theorem. Parseval's inequality. 6. Complex analysis: Elementary functions of complex variable. Limits, continuity and derivability. Holomorphy. Points of singularity and their classification. Integration in the complex field. Cauchy integral theorem and formula. Laurent series and classification of singularity points. Residues and residue theorem.
Methods of in-progress profit verifications The degree of student learning is constantly monitored through verification tools and methodologies. In particular, in order to make the verification and certification of training outcomes feasible, the lecturer and tutor will take into account the following aspects: 1. automatic tracking of training activities by the system - reporting; 2. didactic and technical monitoring (at the level of quantity and quality of interactions, adherence to didactic deadlines, delivery of scheduled papers, etc.); 3. formative verifications in itinere, including for self-assessment (e.g. multiple choice tests, true/false, sequence of questions with different difficulty, simulations, concept maps, papers, group projects, etc.); 4. final proficiency exam, in which not only the performance in the exam is taken into account and valued, but also the work done online (activities carried out at a distance, quantity and quality of online interactions, etc.). Evaluation, within this framework, takes into account several aspects: a. the outcome of a number of intermediate tests, if any (in terms of online tests, development of papers, etc.); b. the quality and quantity of participation in online activities (frequency and quality of interventions that can be monitored through the platform); c. the results of the final test. It is specified that the final test is "written" only. The oral test is optional and is only and exclusively useful to refine the evaluation of the written. The topics of the oral test, if conducted, are agreed upon between the lecturer and the student. All data collected will, in general, be evaluated by the lecturer in order to give the student an assessment that is objective and consistent with the objectives of the university, taking into account both summative and formative aspects
Assessment modalities Access to the exam is subject to the recognition of attendance, which will be sanctioned with the appropriate certificate at the time of booking the exam, which will attest to the performance of the educational activities of verification in itinere and to the level of the work done in the various exercises. The examination will consist of a written test, with optional oral test. The final grade will be expressed in 30ths, with possible honors depending on the degree of maturity achieved by the student. The exam aims at the educational objectives described below. Specifically, the student should: (a) be able to recognize a possible methodology for solving an indefinite, or definite integral; (b) be able to study the character of a number series; (c) be able to identify possible relative maxima/minimum for functions of two variables; (d) be able to solve various types of differential equations; (e) be able to study the convergence of appropriate types of series of functions; (f) be able to solve problems related to Complex Analysis.
Objectives of the test The written test provides a score for all of the above objectives from (a) to (f). Each objective has a variable score. Honors will be awarded in those particular cases where special maturity is demonstrated in solving the examination questions.
Knowledge and ability to understand in terms of expected results (Dublin descriptor No. 1) - Knowledge of the theory on integrals will enable appropriate estimation of the volume of traffic flows in telecommunications networks. - Knowledge of the theory on numerical series will enable the definition of convergence techniques for ad hoc numerical schemes for the simulation of data traffic networks. - Knowledge of the theory on functions of two variables will enable the student to define appropriate optimization techniques for traffic performance in telecommunications networks. - Knowledge of the theory on differential equations will provide a simpler understanding of variational phenomena in the contexts of Information and Communication Technology systems. - Knowledge of the theory on function sets will enable the development of special methodologies for managing the convergence of numerical traffic simulation schemes. - Knowledge of Complex Analysis theory will enable the handling of models and methods for signal analysis.
Knowledge and understanding in terms of expected outcomes (Dublin descriptor No. 2) - The student will be able to use knowledge of integration theory, numerical series, and function series to solve analytical/numerical problems commonly encountered in ordinary traffic networks. - The student will be able to use knowledge about functions of several variables to formulate traffic optimization problems in traffic networks. - The student will be able to use knowledge about differential equations to understand the speed of data variation dynamics in Telecommunication networks. - The student will be able to use knowledge on Complex Analysis to define general characteristics for the design of data transmission systems.
(reference books)
- Paolo Marcellini, Carlo Sbordone, Elements of Mathematical Analysis 2, Simplified version for new undergraduate courses, Liguori Editore, 2016. - Ciro D'Apice, Rosanna Manzo, Toward the Mathematical Analysis 2 exam, Maggioli Editore, 2015. Supplementary teaching materials will be available in the teaching section within the university platform.
|