27210 - Mathematical Analysis 1 (A-L)

Learning outcomes

At the end of the course, the student acquires basic knowledge of infinitesimal and integral calculus, while developing a habit of scientific reasoning and a sensitivity to the analysis of mathematical models, particularly through the study of the asymptotic development of functions. Additionally, the student is able to conduct a detailed study of functions of one variable, as well as sequences and series, both numerical and functional.

Course contents

MATHEMATICAL LOGIC

Brief introduction to logic. Symbols and types of proof.

SETS

Definition of numerical sets N, Z, Q, R. Proofs by induction. Axiomatic construction of the reals. Existence of the lower and upper bounds of a set. Cardinality of a set, countability of rationals, uncountability of real numbers. The power set of a given set has strictly greater cardinality than that of the set.

ELEMENTS OF TOPOLOGY ON THE REAL LINE

Absolute value and its properties. Definition of an open subset. Definition of a closed set. Definition of an accumulation point. Definition of a bounded set. Definition of a compact set. Definition of a connected set.

SEQUENCES AND SERIES

Definition of a convergent sequence. Algebra of limits. Definition of a divergent sequence. Comparison theorem. Monotone and bounded sequences are convergent. Definition of the number e. Cauchy sequences and the second definition of completeness (a sequence in R is convergent if and only if it is a Cauchy sequence). Definition of a convergent series. Necessary condition for the convergence of a series. Series with non-negative terms (ratio test, root test, and condensation test). Geometric series, harmonic series, and generalized harmonic series. Absolute convergence. Leibniz criterion (for alternating series).

FUNCTIONS

Definition (and existence) of the n-th root of a non-negative number. Definition of exponentiation with a real exponent. Continuous functions (definition and main theorems: zero theorem, intermediate value theorem, and Weierstrass theorem). Uniformly continuous functions. Heine-Cantor theorem. Definition of elementary functions (exponentiation with a real exponent, exponential functions, logarithms, trigonometric functions with their inverse functions). Continuity of elementary functions. Continuity of the inverse function of a continuous and invertible function defined on a connected interval.

DERIVATIVES

Definition of derivative. Algebra of derivatives. Derivative of the composite function. Derivative of the inverse function. Differentiability of elementary functions. Rolle's theorem, Lagrange's theorem, Cauchy's theorem, and L'Hopital's theorem. Taylor's formula with Peano, integral, and Lagrange remainders.

INTEGRAL

Primitives. Darboux's theorem. Definition of the Riemann integral of a bounded function. Riemann integrability of continuous (and bounded) functions. Properties of the integral. Integration methods: integration by parts, substitution method, partial fractions (with real or complex roots of arbitrary multiplicity). Mean value theorem and fundamental theorem of calculus. Improper integrals.

SEQUENCES AND SERIES OF FUNCTIONS

Uniform convergence of a sequence of functions. Completeness of the set of continuous functions defined on a compact set with respect to the distance associated with the uniform norm. Limit interchange under the integral sign. Total convergence of a series of continuous functions.

ORDINARY DIFFERENTIAL EQUATIONS

First-order ordinary differential equations with separable variables. First-order linear differential equations. Method of variation of the arbitrary constant. Second-order linear differential equations with constant coefficients (homogeneous and non-homogeneous, i.e., with a "source" term). Solution of the Cauchy problem for the previous classes of equations. n-th order linear differential equations with variable coefficients. Contraction theorem. Existence and uniqueness theorem for solutions of first-order differential equations (with scalar and vector values).

Readings/Bibliography

• E.Giusti. Analisi matematica 1, Bollati Boringhieri Editore.
  
• G.C.Barozzi, G.Dore, E.Obrecht, Elementi di Analisi Matematica 1, Zanichelli.
• P.Marcellini, C.Sbordone. Elementi di Analisi Matematica Uno, Liguori Editore.

• M.Bramanti, C.Pagani, S.Salsa. Matematica. Calcolo infinitesimale e algebra lineare, Zanichelli Editore.

Teaching methods

Theory and exercises during lectures

Assessment methods

Written Exam (mandatory): The exam involves solving exercises on the course topics. The problems cover: function study, calculation of complex numbers, limit calculation, study of the convergence of numerical and functional series, solving integrals, and ordinary differential equations. The theoretical exam can be taken after scoring at least 18 on the written exam.

Theoretical Exam: Consists of a written part and an optional oral part. Details will be explained during lectures.

Teaching tools

Exercices proposed on Virtuale

Office hours

See the website of Berardo Ruffini