B000023 - REAL ANALYSIS

Italian

Italian

Italian

Real and complex numbers
Functions of one variable
Limits
Derivatives
Taylor's formula
Riemann's integral
Improper integrals
Numerical sequences and series.
Elements of point set topology
Functions of several variables
Partial and directional derivatives, continuity and differentiability, free and constrained extrema.
Double and triple integrals
Parametric curves
Line and surface integrals
Ordinary differential equations
Method of separation of variables and linear equations

Real and complex numbers,
Real Functions of a real Variable
Topology of the real line,
Limits,
Continuity,
Derivatives,
Taylor formula,
Riemann integral,
Primitive of a function,
Improper integrals,
Sequences and numerical series,
Power series.
Functions of several variables.
Euclidean spaces R^n and thei topology,
Limits, continuity,
Directional and partial derivatives, gradient, applications,
Double and triple integrals, applications,
Ordinary differential equations,
Line integrals

Textbooks:
Anichini G. - Conti G., Analisi Matematica 1,Pearson Education, 2008.
Anichini G. - Conti G., Analisi Matematica 2,Pearson Education, 2010.

Textbooks recommended for exercises:
Benevieri P., Esercizi di Analisi Matematica, Ed. De Agostini.
Poggiolini L. e Spadini M., Esercizi e temi d'esame di Analisi Matematica II, Esculapio.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 1, Liguori
Editore.
Marcellini P. - Sbordone C., Esercitazioni di Matematica 2, Liguori
Editore.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 1, Zanichelli.
Salsa S. - Squellati A., Esercizi di Analisi Matematica 2, Zanichelli.

Other texts :
Bertsch M. - Dal Passo R. - Giacomelli L., Analisi Matematica, McGraw Hill, Milano 2007.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di una variabile, Pitagora Editrice, Bologna 2005.
Giaquinta M. - Modica G., Note di Analisi Matematica. Funzioni di piu'
variabili, Pitagora Editrice, Bologna 2006.

Se also: http://www.dma.unifi.it/~pera

G. Anichini - G. Conti, Analisi matematica 1, Pearson Ed.

G. Anichini - G. Conti, Analisi Matematica 2, Pearson Ed.

Knowledge and understanding of both mathematical principles and the role of mathematical sciences as a tool for the analysis and the problem solving of mechanical engineering problems. Knowledge of the principles of computer science and the algorithmic and numerical approach to problems.
The training goal is the acquisition of a good disposition to the theoretical approach and to the logical-formal rigor through the elaboration of some concepts of differential and integral calculus in one and several variables, improving students' skills concerning calculation techniques.

Knowledge of mathematical principles and understanding of the role of mathematical sciences as a tool for analysis and resolution of problems and models at the base of industrial engineering and in particular of mechanical engineering. Knowledge of the principles of information technology and of the algorithmic and numerical approach to problems.
The training objective is to acquire a good disposition to the theoretical approach and to the logical-formal rigor through the elaboration of concepts of differential and integral calculus in one or more variables, improving at the same time the familiarity in the calculation.

Basic notions of high school mathematics courses. In particular: formal calculus, polynomials, algebraic equations and inequalities, elements of analytic geometry, and of trigonometry

Fundamentals and basic techniques learned in high school mathematics courses. In particular: formal calculation, polynomials, equations and algebraic inequalities, elements of analytical geometry, trigonometric functions.

The course has two terms consisting in theoretical lectures alternated with tutorials.

It is the first part of an annual course of 12 credits; 6 CFU (Mathematical Analysis 1) and 6 CFU (Mathematical Analysis 2) of frontal teaching consisting of theory lessons and exercises done by the teacher.

More information about the course is available at the following link http://www.dma.unifi.it/~pera

Further information on the course is available on the professor's website

The course has two parts. The exam is performed at the end of the second term. It consists in a written text and in a subsequent oral exam. At the end of the first term an intermediate not mandatory written test is performed.
The student has to develop capability in applying and understanding knowledge related to mathematical methods - with particular reference to differential and integral calculus, geometry, linear algebra, numerical calculation, linear programming, probability and statistics - to model, analyze and solve engineering problems, also with the help of IT tools.

The exam consists in two partial written tests and a subsequent oral exam. The first written exam (Mathematics-module-1) consists of multiple choice problems. Passing this exam is mandatory to access the second written exam. The second written exam (Mathematics-module-2) consists of both multiple choice and open response problems. Access to the oral exam is reserved to students who have passed both the written exams.

The detailed program of the course is available at the following link
http://www.dma.unifi.it/~pera

Numbers: Sets (union, intersection, difference, empty set, complement). Natural, relative, rational numbers.Real numbers: algebraic axioms, sorting. Logical Quantifiers. Inequalities. Absolute value. Powers and roots. Logarithms. Intervals. Maximum, minimum, highest, minor, upper and lower extremes of a set. Properties of real numbers. Density of rational.Applications between sets, injective, suricative, biotic applications. Domain, coding, image and graph of an application.Real functions of a variable (limits and continuity):Real functions of a real variable.Limited funcctions.Monotonous functions.Inverse functions.Polynomials and Rational Functions.Main transcendent functions (exponential and logarithmic functions, trigonometric functions and their inverse, hyperbolic functions).Element of the topology of the real line: neighborhood of a point, acccumulation points, isolated points.Absolute and relative maximum and minimum.Functions limits (endless and infinite).Limit uniqueness theorem.Sign permanence theorem.Theorems for Limit calculation .Left and right limits. Limit existence theorem for monotone functions. Fundamental limits and consequences.Continuity. Continuous theorem of the combined functions (sum, product, quotient and composition). Classification of discontinuities. Zeros theorem. Intermediate values ​​and applications theorem. Continuous theorem of a reverse function.Weierstrass Theorem.Real functions of a variable (derivative): Definition of derivative. Right and left derivatives. Angular points. Geometric Interpretation of Derivatives. Differential.Derivative rules (sum, product, quotient, composition, and inverse function). Derivatives 1 of the main functions.Fermat theorem. Theories of Rolle and Lagrange. Consequences of the Lagrange Theorem. The L'Hopital Theorems.Top Derivatives. Asymptotes of a function. Convex functions within a range.Sufficient conditions for the existence of maximum and minimum levels. Flush points. Function Studies. Infinite and infinite. The o-small symbol.Taylor's formula with the rest in the form of Peano. MacLaurin Formula. Applications of Taylor's Formula to Limit Calculation. Taylor's formula with the rest in the form of Peano.Taylor's formula with the rest in the form of Lagrange (optional) and its applications to some approximation problems.Simple Integers (Primitive):Integral indefinite. Integration formula for parts for indefinite integrals. Integration formula for substitution for indefinite integers. Integration of elementary functions or deduced from elementary functions. Integration of Rational Functions. Some integration methods.Definition of defined integral. Subtle sets and condition necessary and sufficient for integrability. Properties of defined integers (linearity, monotony, additivity).Integration formula by parts for defined integers.Integration formula for substitution for defined integers.Average theorem for integers. Theorem of fundamental calculus.Fundamental formula of integral calculation. Definition of logarithm through integral.Application of the integer defined for the calculation of areas of flat figures and the calculation of volumes of rotation solids.Improper integrals. Convergence criteria (comparison, asymptotic comparison, absolute convergence).The error function.

Function Series. Elements of point set topology. Functions of several variables. Continuity. Partial and directional derivatives. Gradient. Differentiability. Tangent planes. Maximum and minimum. Vector fields. Curl and divergence. Double integrals over rectangles. Double integrals over general regions. Polar coordinates, applications of double integrals. Change of variables. Triple integrals over a rectangular parallelepiped. Triple integrals over general regions. Cylindrical coordinates. Change of variables. Parametric curves. Equations of lines and planes. Vector functions and space curves. Line integrals. Arc length and curvature. Fundamental theorem for line integrals. Ordinary differential equations. Method of separation of variables. Ordinary linear differential equations. Method of variation of parameters.