CODE 52474 ACADEMIC YEAR 2020/2021 CREDITS 12 cfu anno 1 FISICA 8758 (L-30) - GENOVA 16 cfu anno 1 STATISTICA MATEM. E TRATTAM. INFORMATICO DEI DATI 8766 (L-35) - GENOVA 16 cfu anno 1 MATEMATICA 8760 (L-35) - GENOVA SCIENTIFIC DISCIPLINARY SECTOR MAT/05 LANGUAGE Italian TEACHING LOCATION GENOVA PREREQUISITES Propedeuticità in uscita Questo insegnamento è propedeutico per gli insegnamenti: PHYSICS 8758 (coorte 2020/2021) GENERAL PHYSICS 3 57050 PHYSICS 8758 (coorte 2020/2021) ANALYTICAL MECHANICS 25911 PHYSICS 8758 (coorte 2020/2021) MATHEMATICAL ANALYSIS 2 57048 PHYSICS 8758 (coorte 2020/2021) PHYSICS II 57049 MODULES Questo insegnamento è composto da: MATHEMATICAL ANALYSIS 1 MATHEMATICAL ANALYSIS 1 TEACHING MATERIALS AULAWEB OVERVIEW Language: Italian AIMS AND CONTENT LEARNING OUTCOMES Rigorous treatment of Mathematical Analysis, focusing on differential and integral calculus of functions of one real variable. TEACHING METHODS Traditional: blackboard. SYLLABUS/CONTENT 1. Real numbers. The axioms of ordered fields. Absolute value. Natural and integer numbers. Rational numbers and their geometric representation. Completeness and its consequences. Real numbers and the straight line. Archimedean property. Decimal representations. 2. Functions. Relations, functions, domain, codomain, image and graph of a function. Composition of functions. Invertible functions. Operations on real functions. Monotone functions. Polynomials and rational functions. Trigonometric functions. The exponential function on rational numbers. 3. Limits. Metric and e topological properties of R. Continuity. Operations with continuous functions. Limits and their properties. The algebra of limits. Comparison theorems. Limits of monotone functions. Limits of compositions and change of variables. Sequences and their limits. Sunsequences. Bolzano-Weierstrass' theorem. Cauchy sequences. Sequences defined by recurrence and their limits. Neper's number e. 4. Global properties of continuous functions. Weierstrass' theorem. Zeroes of continuous functions. Intermediate value theorem. Continuity and monotonicity. Continuity of the inverse function. Uniform continuity. Heine-Cantor's theorem. The exponential funcion on real numbers. 5. Differential calculus. The derivative: definition and elementary properties. Differentiability and the properties of the differential. Derivative of compositions and inverse functions. Derivatives of elementary functions. Higher order derivatives. The classical theorems by Rolle, Lagrange and Cauchy and their consequences. The theorem of de l'Hopital. Local comparison of functions. Vanishing and diverging functions. Taylor's formula. Convexity. Study of monotonicity and convexity by means of first and second derivatives. Newton's method. Iterative procedures for the solution of equations. 6.The indefinite integral. Integration techniques. Integration of elementary functions. Integration by parts and by substitution. Integration of rational functions. 7. The Riemann integral. Definition and properties of the definite integral. Integrability of continuous and monotonic functions. The oriented integral. The integral mean theorem. Relations between derivation and integration: integral functions, the fundamental theorem of calculus and its consequences. Improper integrals. Convergence criteria. 8. Series. Geometric and telescopic series. Convergence. Series with non negative terms: comparison, root and ratio criteria; condensation, order and integral tests. Alternating series and Leibniz' theorem. 9. Differential equations. Separation of variables. Linear first order equations Second order linear equations with constant coefficients. RECOMMENDED READING/BIBLIOGRAPHY A.Bacciotti, F.Ricci - Analisi Matematica I - Liguori Editore M. Baronti, F. De Mari, R. van der Putten, I. Venturi - Calculus Problems, Springer, 2016 Further readings will be posted on the web page (AULAWEB) TEACHERS AND EXAM BOARD MARCO BARONTI Ricevimento: The teacher is available for explanations one afternoon a week. EMANUELA SASSO ALBERTO SORRENTINO Ricevimento: Friday 8.30-10.30 and on appointment. LESSONS LESSONS START The class will start according to the academic calendar. Class schedule MATHEMATICAL ANALYSIS 1 EXAMS Exam schedule Data appello Orario Luogo Degree type Note Subject 11/01/2021 09:00 GENOVA Scritto 13/01/2021 09:00 GENOVA Orale 01/02/2021 09:00 GENOVA Scritto 03/02/2021 09:00 GENOVA Orale 28/06/2021 09:00 GENOVA Scritto 29/06/2021 09:00 GENOVA Orale 19/07/2021 09:00 GENOVA Scritto 20/07/2021 09:00 GENOVA Orale 03/09/2021 09:00 GENOVA Scritto 07/09/2021 09:00 GENOVA Orale 11/01/2021 09:00 GENOVA Scritto 13/01/2021 09:00 GENOVA Orale 01/02/2021 09:00 GENOVA Scritto 03/02/2021 09:00 GENOVA Orale 28/06/2021 09:00 GENOVA Scritto 29/06/2021 09:00 GENOVA Orale 19/07/2021 09:00 GENOVA Scritto 20/07/2021 09:00 GENOVA Orale 03/09/2021 09:00 GENOVA Scritto 07/09/2021 09:00 GENOVA Orale FURTHER INFORMATION Teaching style: in presence.