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 real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Programme
Part I: Linear algebraThe real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
Core Documentation
Textbook• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Reference Bibliography
Any additional teaching material will be available for students online on the course web page.Type of delivery of the course
Frontal lesson.Type of evaluation
The exam consists in a written test and an optional 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. It is possible to take the exam in English. teacher profile teaching materials
The real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Programme
Part I: Linear algebraThe real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
Core Documentation
Textbook• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Reference Bibliography
Any additional teaching material will be available for students online on the course web page.Type of delivery of the course
Frontal lesson.Type of evaluation
The exam consists in a written test and an optional 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. It is possible to take the exam in English. teacher profile teaching materials
The real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Programme
Part I: Linear algebraThe real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
Core Documentation
Textbook• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Reference Bibliography
Any additional teaching material will be available for students online on the course web page.Type of delivery of the course
Frontal lesson.Type of evaluation
The exam consists in a written test and an optional 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. It is possible to take the exam in English. teacher profile teaching materials
The real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Programme
Part I: Linear algebraThe real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
Core Documentation
Textbook• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Reference Bibliography
Any additional teaching material will be available for students online on the course web page.Type of delivery of the course
Frontal lesson.Type of evaluation
The exam consists in a written test and an optional 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. It is possible to take the exam in English. teacher profile teaching materials
The real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Programme
Part I: Linear algebraThe real vector space . Linear dependence and independence. Subspaces and spaces generated by vectors. Bases. Algebra of matrices. Determinant. Inverse matrix. Rank of a matrix. Systems of linear equations. Cramer's Theorem and Rouché-Capelli's Theorem. Homogeneous systems and linear dependence. Norm and Euclidean distance in . Scalar product. Sequences in . Topology and metrics in . Eigenvalues and eigenvectors. Diagonalization of matrices. Properties of eigenvalues.
Part II: Calculus of functions of several variables:
Continuous functions and Weierstrass theorem. Concave or convex functions. Partial derivatives and gradient. Differential. Derivations along a curve and directional derivative. Second order derivatives. Schwarz's theorem. Hessian matrix. Implicit function thoerem. Comparative statics. Inverse function theorem. Quadratic forms and matrices. Sign of a quadratic form.
Part III: Optimization:
Local and global maximum or minimum. First order conditions and second order conditions for free optimization. Equality constraints. Inequality constraints. Substitution method. Lagrange multiplier theorem. Second order conditions with constraints. Optimization for convex functions. Economic applications.
Part IV: Ordinary differential equations and systems
Definitions and examples. Exact differential. Equations in separable variables. Exact equations. Homogeneous equations. Malthusian growth model. Logistic growth model. Second order linear differential equations. General theorem of existence and uniqueness of the solution. Field of directions. Economic applications. Two-dimensional differential equation systems. Systems of linear differential equations: resolving method using eigenvalues, stationary states and their stability. Economic applications.
Core Documentation
Textbook• Simon & Blume: “Matematica per le scienze economiche” ed. Egea
or
• Simon & Blume: "Matematica 2 per l’economia e le scienze sociali", Università Bocconi Editore, 2002.
Reference Bibliography
Any additional teaching material will be available for students online on the course web page.Type of delivery of the course
Frontal lesson.Type of evaluation
The exam consists in a written test and an optional 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. It is possible to take the exam in English.