The aim of the course is to broaden and consolidate the acquisition of the mathematical method as a fundamental investigation tool for economic, financial and business disciplines. To this end, the course is divided into four parts. In the first part we provide notions of linear algebra to study linear systems and diagonalization of matrices. In the second part the student will be introduced to the study of differential calculus for functions of several variables. In the third part the course provides tools to recognize and study free and constrained optimization problems. In the fourth part solution methods for simple systems and differential equations will be provided.
Curriculum
teacher profile teaching materials
The n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Programme
Part I: Linear algebraThe n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Core Documentation
Simon & Blume: “Matematica per le scienze economiche” ed. Egea.Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Reference Bibliography
Any additional teaching material will be available for students online on the Moodle course web page.Type of delivery of the course
Frontal lesson. Homeworks. Use of graphics tablet and recordings of lectures available at the end of the course.Type of evaluation
The exam consists in a written and an oral test. The written test will consist of exercises or theoretical questions concerning the entire program of the course including poofs of the results indicated in the program with "(w.p.)". The oral test will consist of one or more questions about the entire program. The course will include intermediate tests. It is possible to take the exam in English. teacher profile teaching materials
The n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Programme
Part I: Linear algebraThe n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Core Documentation
Simon & Blume: “Matematica per le scienze economiche” ed. Egea.Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Reference Bibliography
Any additional teaching material will be available for students online on the Moodle course web page.Type of delivery of the course
Frontal lesson. Homeworks. Use of graphics tablet and recordings of lectures available at the end of the course.Type of evaluation
The exam consists in a written and an oral test. The written test will consist of exercises or theoretical questions concerning the entire program of the course including poofs of the results indicated in the program with "(w.p.)". The oral test will consist of one or more questions about the entire program. The course will include intermediate tests. It is possible to take the exam in English. teacher profile teaching materials
The n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Programme
Part I: Linear algebraThe n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Core Documentation
Simon & Blume: “Matematica per le scienze economiche” ed. Egea.Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Reference Bibliography
Any additional teaching material will be available for students online on the Moodle course web page.Type of delivery of the course
Frontal lesson. Homeworks. Use of graphics tablet and recordings of lectures available at the end of the course.Type of evaluation
The exam consists in a written and an oral test. The written test will consist of exercises or theoretical questions concerning the entire program of the course including poofs of the results indicated in the program with "(w.p.)". The oral test will consist of one or more questions about the entire program. The course will include intermediate tests. It is possible to take the exam in English. teacher profile teaching materials
The n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Programme
Part I: Linear algebraThe n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Core Documentation
Simon & Blume: “Matematica per le scienze economiche” ed. Egea.Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Reference Bibliography
Any additional teaching material will be available for students online on the Moodle course web page.Type of delivery of the course
Frontal lesson. Homeworks. Use of graphics tablet and recordings of lectures available at the end of the course.Type of evaluation
The exam consists in a written and an oral test. The written test will consist of exercises or theoretical questions concerning the entire program of the course including poofs of the results indicated in the program with "(w.p.)". The oral test will consist of one or more questions about the entire program. The course will include intermediate tests. It is possible to take the exam in English. teacher profile teaching materials
The n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Programme
Part I: Linear algebraThe n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Core Documentation
Simon & Blume: “Matematica per le scienze economiche” ed. Egea.Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Reference Bibliography
Any additional teaching material will be available for students online on the Moodle course web page.Type of delivery of the course
Frontal lesson. Homeworks. Use of graphics tablet and recordings of lectures available at the end of the course.Type of evaluation
The exam consists in a written and an oral test. The written test will consist of exercises or theoretical questions concerning the entire program of the course including poofs of the results indicated in the program with "(w.p.)". The oral test will consist of one or more questions about the entire program. The course will include intermediate tests. It is possible to take the exam in English. teacher profile teaching materials
The n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Programme
Part I: Linear algebraThe n-dimensional real vector space. Euclidean norm and distance in n-dimensional real vector space. Scalar product of vectors. Topology and metrics in n-dimensional real vector spaces. Linear dependence and independence. Bases. Subspaces and spaces generated by vectors. Eigenvalues and eigenvectors. Diagonalisation of matrices. Properties of eigenvalues.
Part II: Functions of several variables - Differential calculus and free optimization
Functions defined between Euclidean spaces. Graphs and level curves. Linear functions and representation theorem. Quadratic functions and representation theorem. Continuous functions and Weierstrass Theorem. Concave and convex functions. Partial derivatives and gradient. Differential. Derivative along a curve and directional derivative. First property of the gradient (w.p.). Higher order derivatives. Schwarz Theorem. Hessian matrix. Sign of a quadratic form. LPM Criterion. PM Criterion. Definition of local and global maximum or minimum. Free optimization: first-order necessary conditions (w.p.) and second-order sufficient conditions for the existence of local maxima and minima, remarks on the global case under the assumption of concavity/convexity.
Part III: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part IV: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)
Core Documentation
Simon & Blume: “Matematica per le scienze economiche” ed. Egea.Other materials will be available in the course Moodle class.
Other Textbooks:
Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.
Reference Bibliography
Any additional teaching material will be available for students online on the Moodle course web page.Type of delivery of the course
Frontal lesson. Homeworks. Use of graphics tablet and recordings of lectures available at the end of the course.Type of evaluation
The exam consists in a written and an oral test. The written test will consist of exercises or theoretical questions concerning the entire program of the course including poofs of the results indicated in the program with "(w.p.)". The oral test will consist of one or more questions about the entire program. The course will include intermediate tests. It is possible to take the exam in English.