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Program

EN IT

Updated A.Y. 2022-2023

Calculus

- Integral Calculus. Definite and indefinite integrals, Integral properties, The fundamental theorem of calculus, Integration by parts and integration by substitution, Improper integrals.

Linear Algebra

- Linear Spaces. The algebra of vectors, Euclidean Spaces, Inner product, Linear independence.

- Matrices. Matrix algebra, Determinant, Inverse Matrix. 

- Systems of Linear equations. The Gauss Elimination Algorithm, Rouche Capelli theorem.

- Eigenvalues and eigenvectors, Diagonalization.

Optimization

- Calculus of several variables. Domain, Partial derivatives, gradient, hessian matrix. Stationary points. Countour curves.

- Unconstrained optimization. First and second order conditions

- Constrained optimization with Equality constraints. First and second order conditions

- Optimization on a set

 

Suggested Books:

Knut Sydsaeter, Peter Hammond and Arne Strom, Essential Mathematics for Economic Analysis

Carl P. Simon, Lawrence Blume, Mathematics for Economists

Teaching Material

Available on the webpage of the course and on Teams:

  • Slides of the course
  • Additional Exercises

The final Syllabus will be available only at the end of the course. 

You can find below a detailed program which will be update on a daily basis during the teaching period.

 

Lecture 1 (31/10/2022): Definition of atiderivative and integral of a function. Integrals of elementary functions

Lecture 2 (2/11/2022): Integration by substitution and Integration by parts

Practice 1 (3/11/2022): Integrals (elementary integrals, integration by substitution and integration by parts)

Lecture 3 (4/11/2022): Exercises on Integration by parts and Substitution. Construction of the Riemann Integral

Lecture 4 (7/11/2022): Properties of Riemann Integral, Definite integral, Fundamental Theorem of Calculus (FTC). Applications of the FTC.

Lecture 5 (8/11/2022): Exercises on definite integrals, application of FTC, computation of areas. Improper Integrals. Mean Value Theorem.

Lecture 6 (9/11/2022): 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. 

Practice 2 (10/11/2022): Definite Integrals, applications of FTC, Computation of areas. Matrices. 

Lecture 7 (14/11/2022): Matrix-vector product, product between matrices, Properties of the matrix product. Matrix product is not commutative: examples. Definition of the inverse of a matrix. 

Lecture 8 (15/11/2022): 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. 

Lecture 9 (16/11/2022): 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 liunear 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. 

Practice 3: Product of matrices, determinant of a matrix, Inverse of a matrix, linear systems. 

Lecture 10 (21/11/2022): Exercises on linear systems, linear dependence and linear independence of vectors. 

Lecture 11 (22/11/2022): Linear independence of vectors, rank of a matrix. Methods for computing the rank of a matrix: Gauss Elimination algorithm and Kronecker Theorem. 

Lecture 12 (23/11/2022): Exercises on the computation of the rank. 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.  

Practice 4 (24/11/2022): Linearly dependent/independent vectors, rank, parametric systems

Lecture 13 (28/11/2022): 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.  

Lecture 14 (29/11/2022): Eigenvalues and Eigenvectors computation; functions of two variables: domain. 

Lecture 15 (30/11/2022): Functions of two variables: level curves, partial derivatives, gradient.

Practice 5 (1/12/2022): Parametric/cartesian equations of lines/planes. Eigenvalues and Eigenvectors. Functions of two variables: domain, level curves, gradient. 

Lecture 16 (5/12/2022): Partial derivatives, gradient, Stationary points. Hessian matrix, concavity, convexity.  

Lecture 17 (6/12/2022): Tangent plane, Optimization of functions of two variables. 

Lecture 18 (7/12/2022): Again on Optimization of functions of two variables.

Practice 6 (7/12/2022): Tangent plane and Optimization of functios of two variables.