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MATHEMATICAL ANALYSIS I

CODE 56594
ACADEMIC YEAR 2021/2022
CREDITS
  • 12 cfu during the 1st year of 10375 INGEGNERIA CHIMICA E DI PROCESSO (L-9) - GENOVA
  • 12 cfu during the 1st year of 9273 INGEGNERIA ELETTRONICA E TECNOLOGIE DELL'INFORMAZIONE(L-8) - GENOVA
  • 12 cfu during the 1st year of 8716 INGEGNERIA ELETTRICA (L-9) - GENOVA
  • SCIENTIFIC DISCIPLINARY SECTOR MAT/05
    LANGUAGE Italian
    TEACHING LOCATION
  • GENOVA
  • SEMESTER Annual
    PREREQUISITES
    Prerequisites (for future units)
    This unit is a prerequisite for:
    • Chemical Engineering 8714 (coorte 2021/2022)
    • MATHEMATICAL ANALYSIS II 60241
    • ELECTRICAL ENGINEERING 66016
    • THEORY OF DEVELOPMENT OF CHEMICAL PROCESSES 66364
    • TRAINING AND ORIENTATION 66376
    • SCIENCE AND TECHNOLOGIES OF MATERIALS 84498
    • Electrical Engineering 8716 (coorte 2021/2022)
    • FOUNDATIONS OF ELECTRICAL ENGINEERING 60334
    • POWER GENERATION 60221
    • STRUCTURAL MECHANICS 66283
    • ELECTRONICS FOR ELECTRICAL ENGINEERING 84372
    • MATHEMATICAL PHYSICS 1 60352
    • ELECTRIC AND MAGNETIC FIELDS 60335
    • CIRCUIT THEORY 60336
    • MATHEMATICAL ANALYSIS II 60241
    • APPLIED PHYSICS 60359
    • SOLID AND MACHINE MECHANICS 80338
    • MECHANICS OF MACHINES 86899
    • Electronic Engineering and Information Technology 9273 (coorte 2021/2022)
    • MATHEMATICAL METHODS FOR ENGINEERING 72440
    • CHEMICAL AND PROCESSES ENGINEERING 10375 (coorte 2021/2022)
    • SCIENCE AND TECHNOLOGIES OF MATERIALS 84498
    • MATHEMATICAL ANALYSIS II 60243
    • CHEMICAL ENGINEERING LABORATORIES 90664
    • CHEMICAL AND PROCESS PLANTS 90660
    • STRUCTURAL MECHANICS 90682
    • CHEMICAL REACTORS 90669
    • THEORY OF DEVELOPMENT OF CHEMICAL PROCESSES 66364
    • MATHEMATICAL ANALYSIS II AND PHYSICS 90657
    TEACHING MATERIALS AULAWEB

    OVERVIEW

    The course "Mathematical Analysis I" aims to provide students with some basic mathematical tools, both theoretical and computational, useful for engineering and application-oriented topics of all the next courses.

    The course will be focused on functions of one and several real variables, on the related differential and integral calculus, on the resolution of ordinary differential equations and series of functions.

     

    AIMS AND CONTENT

    LEARNING OUTCOMES

    The course introduces general mathematical notions and tools at the basis of engineering modeling, related to the study of the functions of one or more real variables. In particular, the concept of limit and continuity, the differential and integral calculus, also of functions of several real variables, the resolution of ordinary differential equations, the analysis of curves and surfaces, and the study of the convergence of numerical series and series of functions.

    AIMS AND LEARNING OUTCOMES

    The "Mathematical Analysis I" course aims at giving basic mathematical tools necessary to the studies in the engineering field.

    At the end of the lessons the student will have acquired sufficient theoretical knowledge:

    • to identify, understand, formulate and solve general engineering problems related to mathematically modeled quantitie using appropriate analytical methods;
    • to combine notions of theory into practice to solve basic model engineering problems;
    • to be able to autonomously learning new mathematical tools useful for engineering applications, throughout the entire working life;
    • to analyze and model geometric and physical objects related to functions of one or more real variables, and to calculate quantities associated with them;
    • to apply mathematical resolution tools in the context of the differential calculation of the functions of one or more real variables;
    • to apply mathematical resolution tools in the context of the integral calculation of the functions of one real variable;
    • to compute the maximum and minimum unconstrained values ​​of functions of one and several variables, useful in application areas of optimization;
    • to analyze and model geometric objects related to curves, and calculate associated quantities;
    • to understand and solve simple models related to ordinary differential equations, through which physical phenomena of engineering interest are represented;
    • to know the concept of numerical series and series of functions and to evaluate their convergence, useful in the approximate calculation of quantities in the numerical-computational field.

    PREREQUISITES

    Elementary algebra: literal calculus, polynomials, equations and inequalities, trigonometry.

    TEACHING METHODS

    72 hours of theoretical lessons, 48 hours of classroom practices. During the theoretical lessons the definitions and the theorems will be presented with many examples and applications. During the other part of the course many exercises will be solved. 

    In addition, a tutor will solve some exercises in extra (optional) lesson hours.

    SYLLABUS/CONTENT

    The teaching program includes both theoretical study and practical resolution of exercises in the following topics:

    • Sets, logic, real numbers, infimum and supremum
    • Functions of one real variable, elementary functions, limits, infinitesimals and infinities, continuous functions, derivable functions, differentiable functions. Taylor’s formula, expansion of elementary functions.
    • Primitives and indefinite integrals, methods of indefinite integration, definite integrals, fundamental theorem of integral calculus.
    • Functions of several variables (scalar and vectorial fields), limits and continuity. Directional derivatives. Differentiable functions. Necessary and sufficient conditions for differentiability. Derivatives of composite functions. Derivatives of higher order, Schwarz Theorem and Taylor polynomial in several variables. Unconstrained maxima and minima of scalar fields, necessary and sufficient conditions, Hessian matrix.
    • Differential equations of the first order, with separable variables, linear, homogeneous, Bernoulli and Riccati types. Existence and uniqueness theorem (hints) for the Cauchy’s problem. Linear differential equations. Linear differential equations of higher order with constant coefficients, homogeneous and non-homogeneous,
    • Improper integrals of one variable.
    • Numerical series. Convergence criteria for constant sign numerical series. Numerical alternating series and absolutely convergent series. Series of functions. Pointwise, absolute, uniform and total convergence criteria. Derivation and integration of several functions (hints). Power series, radius of convergence.

    RECOMMENDED READING/BIBLIOGRAPHY

    Handouts: "MATHEMATICS I"  and "MATHEMATICS II" by prof. Maurizio Romeo, downloadable for free from the web page of the course.

    Sheets containing links to web pages with different solved exercises, downloadable for free from the web page of the course.

    Workbook: Laura Recine - Maurizio Romeo, Esercizi di analisi matematica - Volume II, Maggioli Editore.

    P. Marcellini – C. Sbordone: Calcolo, Liguori Editore, Napoli, or any other good text of mathematical analysis.

    M.Baronti – F.De Mari – R.Van Der Putten – I.Venturi: Calculus Problems, Springer

     

    TEACHERS AND EXAM BOARD

    Exam Board

    CLAUDIO ESTATICO (President)

    MARCO BARONTI (President Substitute)

    ULDERICO FUGACCI (President Substitute)

    LESSONS

    EXAMS

    EXAM DESCRIPTION

    The final exam consists of a written test and an oral exam. The student must obtain an evaluation of at least 16/30 in the written test to access the oral exam.

    Students with SLD other special educational needs are advised to contact the teachers at the beginning of the course to agree on the most appropriate teaching and exam methods that, in compliance with the teaching objectives, meet the individual needs and to identify student-specific compensatory tools.

    ASSESSMENT METHODS

    The exam consists of a written test and an oral test.

    The written test consists in solving exercises concerning the arguments of the course. The written test must be passed before attending the oral examination and can be taken both in previous sessions and in the same session in which the student intends to attend the oral examination.

    Only students who have previously passed the written test with a grade greater than or equal to 16/30 can access the oral exam.

    Exam schedule

    Date Time Location Type Notes
    17/01/2022 09:00 GENOVA Scritto
    24/01/2022 09:00 GENOVA Orale
    07/02/2022 09:00 GENOVA Scritto
    16/02/2022 09:00 GENOVA Orale
    15/06/2022 09:00 GENOVA Scritto
    20/06/2022 09:00 GENOVA Orale
    15/07/2022 09:00 GENOVA Scritto
    20/07/2022 09:00 GENOVA Orale
    09/09/2022 09:00 GENOVA Scritto
    15/09/2022 09:00 GENOVA Orale

    FURTHER INFORMATION

    Attendance is not compulsory but strongly recommended to all students.