The course provides the bases of mathematical analysis which are more frequently employed in statistics and insurance.

KNOWLEDGE AND UNDERSTANDING

Knowing differentiability for multivariable functions, real and function sequences and series and integration theory.
Understanding the main tecniques usually employed in these frameworks.
Understanding some applications of these topics to the calculus.

APPLYING KNOWLEDGE AND UNDERSTANDING

Employing results and theorems presented in the course to solve some computing problems.
Using results and theorems presented in the course to prove some theoretical statements.

MAKING JUDGEMENTS

Evaluating which tools should be preferably employed in solving some computing problems.
Choosing the most appropriate tools to prove some theoretical results.

COMMUNICATION SKILLS

Explaining why some tools should be preferably chosen in solving some computing problems or proving some theoretical results.

LEARNING SKILLS

Employing the knowledge acquired to learn more advanced or complementary mathematical topics.

A good mastery in the topics presented in Mathematics for Economics and Statistics I and II is required in order to have a good understanding of the content of the course. Passing the exam of the preliminary course 'Mathematics for Economics and Statistics I' is mandatory to sit for the exam of this course.

a) Real sequences and series: main theorems and convergence criteria. Cauchy sequences. Sequences and series of functions. Pointwise and uniform convergence. Theorem on the exchage of the order of limits and its consequences. Uniform convergence and its relation with continuity, differentiability and integrals. Weirstrass criterion on the uniform convergence of functions series. Power series in R; convergence radius and how to computer it. Infinitely often differentiability of a power series. Taylor series. Expression of some elementary functions as power series. Examples and applications of criteria and theorems. b) Riemann-Stieltjes integral. c) Multivariable calculus and differentiability. Some hints on the geometry in the n-dimensional Euclidean space: scalar product, distance, Cauchy inequality, norm, some hints on topology; polar coordinate system in RxR. Limites and continuity of multivariable functions. Connected and compact sets and their relationship with continuity; Derivability and differentiability for multivariable functions. Jacobian matrix and gradient. Differentiability of composite functions. Derivability and differentiability of higher order. Schwarz theorem. Taylor formula. Implicit functions. d) Optimisation of multivariate functions. Unconstrained extreme points for multivariable functions. Necessary and sufficient conditions. Hessian matrix. Quadratic forms. Constrained extreme points; method of Lagrange multipliers. Convex functions. Kuhn-Tucker Theorem. e) Multiple Riemann integral. Definition of doulbe integral for functions defined on rectangles; theorem on reducing double integrals. Integral of functions defined on a more general domain. Peano-Jordan measure. Integral of functions with finitely many discontinuities. Main cases of integrability. Reducing formulae. Multiple integrals and reducing formulae for multivariable functions. Changing variables in multiple integrals. Computing volumes by means of integrals.

Carlo Domenico Pagani, Sandro Salsa
Analisi matematica - vol. 1 - 2015 - ISBN: 9788808151339
Analisi matematica - vol. 2 - 2016 - ISBN: 9788808637086

Sandro Salsa, Annamaria Squellati
Esercizi di Analisi matematica – volume 1 - 2011 - ISBN: 9788808218940
Esercizi di Analisi matematica – volume 2 - 2011 - ISBN: 9788808218964

Information about the sections of the books that will be employed in the course will be given during the lectures and will also be indicated in the Moodle website, where further study material will be available (slides).

a) Real sequences and series: main theorems and convergence criteria. Cauchy sequences. Sequences and series of functions. Pointwise and uniform convergence. Theorem on the exchage of the order of limits and its consequences. Uniform convergence and its relation with continuity, differentiability and integrals. Weirstrass criterion on the uniform convergence of functions series. Power series in R; convergence radius and how to computer it. Infinitely often differentiability of a power series. Taylor series. Expression of some elementary functions as power series. Examples and applications of criteria and theorems. b) Riemann-Stieltjes integral. c) Multivariable calculus and differentiability. Some hints on the geometry in the n-dimensional Euclidean space: scalar product, distance, Cauchy inequality, norm, some hints on topology; polar coordinate system in RxR. Limites and continuity of multivariable functions. Connected and compact sets and their relationship with continuity; Derivability and differentiability for multivariable functions. Jacobian matrix and gradient. Differentiability of composite functions. Derivability and differentiability of higher order. Schwarz theorem. Taylor formula. Implicit functions. d) Optimisation of multivariate functions. Unconstrained extreme points for multivariable functions. Necessary and sufficient conditions. Hessian matrix. Quadratic forms. Constrained extreme points; method of Lagrange multipliers. Convex functions. Kuhn-Tucker Theorem. e) Multiple Riemann integral. Definition of doulbe integral for functions defined on rectangles; theorem on reducing double integrals. Integral of functions defined on a more general domain. Peano-Jordan measure. Integral of functions with finitely many discontinuities. Main cases of integrability. Reducing formulae. Multiple integrals and reducing formulae for multivariable functions. Changing variables in multiple integrals. Computing volumes by means of integrals.

Lectures. Many examples and exercises will be presented to explain and clarify the theoretical results presented in the course.

The detailed course program will be available in the Moodle website.

Students are required to be prepared on the whole content of the course. Besides, they are required not only to memorise but to have a good comprehension of the course content as well.
The exam consists in a written test, followed by an oral test. The latter is subject to admission. To be admitted to the oral test the mark received in the written test must be at least 16/30. The written test requires solving some exercises and, possibly, proving some simple theoretical results. The oral test consists in a discussion on the topics presented in the course. Proving some theorems or other results, explaining and applying them and also solving some exercises might be requested as well.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs)