Syllabus
EN
IT
Learning Objectives
LEARNING OUTCOMES:
Knowledge and understanding of the basic principles useful for a deep comprehension of the main concepts of mathematical analysis, particularly related to sequences, series and functions of one variable. The objective is to apply basic notions on mathematical analysis to economic modelling with a rigorous approach.
KNOWLEDGE AND UNDERSTANDING:
The course develops basic topics of mathematical analysis. It provides basic knowledge of economic modeling and financial mathematics.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The structure of the course consists of a theoretical part (definitions and theorems with proofs), the discussion of useful examples for economic theory, supplemented by fundamental practical exercises in which concrete methods are developed for the solution of basic problems in mathematical analysis and linear algebra. The understanding of proofs allows students to have the right approach for facing different types of problems of moderate difficulty, in the field of economic modeling and in the understanding, based on quantitative methods, of social phenomena.
MAKING JUDGEMENTS:
The course includes proofs of theorems and analytic properties. This aspect allows students to construct and develop logical arguments with a clear identification of assumptions and conclusions, to recognize correct proofs, and to identify incorrect or incomplete reasoning. Various examples of applications to economic models are also studied, which enable students to propose and analyze mathematical models useful for social sciences.
COMMUNICATION SKILLS:
To succeed the written exam the student must develop the necessary scientific rigor required to describe the analytical solution of a problem. The course therefore provides some tools needed to communicate, rigorously, scientific results in the social sciences.
LEARNING SKILLS:
The course provides basic tools for the development of further studies, both in Mathematics and in Economics. The theoretical study provides the ability to deal independently with new problems of medium difficulty.
Knowledge and understanding of the basic principles useful for a deep comprehension of the main concepts of mathematical analysis, particularly related to sequences, series and functions of one variable. The objective is to apply basic notions on mathematical analysis to economic modelling with a rigorous approach.
KNOWLEDGE AND UNDERSTANDING:
The course develops basic topics of mathematical analysis. It provides basic knowledge of economic modeling and financial mathematics.
APPLYING KNOWLEDGE AND UNDERSTANDING:
The structure of the course consists of a theoretical part (definitions and theorems with proofs), the discussion of useful examples for economic theory, supplemented by fundamental practical exercises in which concrete methods are developed for the solution of basic problems in mathematical analysis and linear algebra. The understanding of proofs allows students to have the right approach for facing different types of problems of moderate difficulty, in the field of economic modeling and in the understanding, based on quantitative methods, of social phenomena.
MAKING JUDGEMENTS:
The course includes proofs of theorems and analytic properties. This aspect allows students to construct and develop logical arguments with a clear identification of assumptions and conclusions, to recognize correct proofs, and to identify incorrect or incomplete reasoning. Various examples of applications to economic models are also studied, which enable students to propose and analyze mathematical models useful for social sciences.
COMMUNICATION SKILLS:
To succeed the written exam the student must develop the necessary scientific rigor required to describe the analytical solution of a problem. The course therefore provides some tools needed to communicate, rigorously, scientific results in the social sciences.
LEARNING SKILLS:
The course provides basic tools for the development of further studies, both in Mathematics and in Economics. The theoretical study provides the ability to deal independently with new problems of medium difficulty.
Prerequisites
All the material covered in Mathematics 1
Program
Integral Calculus
Definition of antiderivative and integral of a function.
Integrals of elementary functions.
Integration by substitution, Integration by parts, integrals of ratio of polynomials.
Construction of the Riemann Integral. Properties of Riemann Integral, Definite integral, Fundamental Theorem of Calculus (FTC) with proof. Applications of the FTC.
Computation of areas. Improper Integrals. Mean Value Theorem.
Linear Algebra
Matrices and Vectors: definitions, operations with matrices and vectors (sum, subtraction, multiplication by a real number, transpose, inner product). Geometric interpretation of vectors. Linear combination of vectors. Any vector in R^n is a linear combination of standard vectors (with proof).
Matrix-vector product, product between matrices, Properties of the matrix product. Matrix product is not commutative: examples. Definition of the inverse of a matrix.
Theorem: the inverse of a matrix is unique (with proof). Inverse of a 2x2 matrix. Determinant of a matrix (cofactor expansion by row and by column). Properties of determinants. Theorem: A matrix is invertible iff its determinant is not equal to zero. Properties of the inverse of a matrix (with proof). The cofactor matrix. Computation of the inverse of a matrix. Special matrices: Diagonal and Triangular matrices.
Definition of a linear equation and a system of linear equations. Matrix-vector expression for a system of linear equations. Definition of the solution of a system of linear equations. Consistent and Inconsisten systems. Theorem: A homogeneous system is always consistent (with proof). Elementary Row operations and the Gauss Elimination Algorithm. Row Echelon Form, consistency and inconsistency of a system.
Linear dependence and linear independence of vectors.
Rank of a matrix. Methods for computing the rank of a matrix: Gauss Elimination algorithm and Kronecker Theorem.
Proposition: Given a set of vectors v_1, ..., v_n in R^k and a vector b in R^k which is a linear combination of vectors v_1, ..., v_n, then the matrix V and the matrix tilde V have the same rank. (with proof). Rouché-Capelli Theorem (Statement and interpretation). Parametric systems.
Geometric interpretation of the solutions of a linear system; cartesian and parametric equation of a line through two points, and through a point in a direction v; parametric equation of a plane through a point and parallelt to two vectors, point-normal equation of a plane in R^3. Eigenvalues and Eigenvectors definition, computation.
Optimization
Functions of two variables: domain, level curves, partial derivatives, gradient. Stationary points. Hessian matrix, concavity, convexity. Tangent plane, Optimization of functions of two variables.
Definition of antiderivative and integral of a function.
Integrals of elementary functions.
Integration by substitution, Integration by parts, integrals of ratio of polynomials.
Construction of the Riemann Integral. Properties of Riemann Integral, Definite integral, Fundamental Theorem of Calculus (FTC) with proof. Applications of the FTC.
Computation of areas. Improper Integrals. Mean Value Theorem.
Linear Algebra
Matrices and Vectors: definitions, operations with matrices and vectors (sum, subtraction, multiplication by a real number, transpose, inner product). Geometric interpretation of vectors. Linear combination of vectors. Any vector in R^n is a linear combination of standard vectors (with proof).
Matrix-vector product, product between matrices, Properties of the matrix product. Matrix product is not commutative: examples. Definition of the inverse of a matrix.
Theorem: the inverse of a matrix is unique (with proof). Inverse of a 2x2 matrix. Determinant of a matrix (cofactor expansion by row and by column). Properties of determinants. Theorem: A matrix is invertible iff its determinant is not equal to zero. Properties of the inverse of a matrix (with proof). The cofactor matrix. Computation of the inverse of a matrix. Special matrices: Diagonal and Triangular matrices.
Definition of a linear equation and a system of linear equations. Matrix-vector expression for a system of linear equations. Definition of the solution of a system of linear equations. Consistent and Inconsisten systems. Theorem: A homogeneous system is always consistent (with proof). Elementary Row operations and the Gauss Elimination Algorithm. Row Echelon Form, consistency and inconsistency of a system.
Linear dependence and linear independence of vectors.
Rank of a matrix. Methods for computing the rank of a matrix: Gauss Elimination algorithm and Kronecker Theorem.
Proposition: Given a set of vectors v_1, ..., v_n in R^k and a vector b in R^k which is a linear combination of vectors v_1, ..., v_n, then the matrix V and the matrix tilde V have the same rank. (with proof). Rouché-Capelli Theorem (Statement and interpretation). Parametric systems.
Geometric interpretation of the solutions of a linear system; cartesian and parametric equation of a line through two points, and through a point in a direction v; parametric equation of a plane through a point and parallelt to two vectors, point-normal equation of a plane in R^3. Eigenvalues and Eigenvectors definition, computation.
Optimization
Functions of two variables: domain, level curves, partial derivatives, gradient. Stationary points. Hessian matrix, concavity, convexity. Tangent plane, Optimization of functions of two variables.
Books
Lorenzo Peccati, Sandro Salsa, Annamaria Squellati. Mathematics for economic business.
Bibliography
Knut Sydsæter, Peter Hammond, Arne StrØm, Essential Mathematics for Economic Analysis
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Carl P. Simon, Lawrence Blume, Mathematics for Economists
Teaching methods
Theoretical lectures and exercise classes
Exam Rules
Mathematics is a single corse split in two modules. After the final exam only one grade will be registered.
The exam consists of a set of exercises and theoretical questions on the material covered during the course.
The exam is passed if the mark is grater than or equal to 18/30.
Every result of the exam will be registered, including "fail". Students are allowed to reject a mark greater or equal to 18 only once and re-take the exam. In the case, at the second time the grade will be registered.
The exam consists of a set of exercises and theoretical questions on the material covered during the course.
The exam is passed if the mark is grater than or equal to 18/30.
Every result of the exam will be registered, including "fail". Students are allowed to reject a mark greater or equal to 18 only once and re-take the exam. In the case, at the second time the grade will be registered.
Attendance Rules
Attendance is not compulsory but highly recommended