Teaching

Description of the Course

E-learning platform (students only)

The aim of the course is to provide a deepest introduction to the basic tools and theorems of mathematical calculus. Standard courses for students in economics are oriented to provide a specific set of computational procedures to be applied to the solution of specific problems. The present course might constitute a good complement as it is more concerned with the definition and study of general notions and it is characterized by a theorem-proving approach.

• Teaching load: 40 hours
• Lecturer : BOTTAZZI Giulio
• Teaching Assistant: TBD
• Semester: Fall
• Textbook: Advanced Calculus for Economics and Finance
• Further references and Optional readings: Principles of Mathematical Analysis, W.A. Rudin, chapters 3, 6 and 9
• Prerequisites: Basics of topology, linear algebra, metric spaces, and sequences and series.

List of topics covered by the lectures (not necessarily in this order)

Normed Spaces:

• Definition and Basic Properties
• Euclidean Norm in Rn
• p-Norm in Rn
• Operator Norm
• Finite-Dimensional Normed Spaces
• Equivalence of Norms in Rn

Differential calculus of real functions:

• Properties of derivatives.
• Continuity and differential.
• Lagrange theorem.
• de L'Hopital rule and application.
• Derivatives of fundamental functions.

Differential calculus of functions of many variables:

• Limits and Continuity in Rn
• Differential Analysis in Rn
• Mean Value Theorems
• Higher-Order Derivatives and Taylor Polynomial
• Local Maxima and Minima
• Inverse Function Theorem
• Implicit Function Theorem
• Constrained Optimisation

Constrained optimisation:

• Theorems of alternatives
• First-order conditions and Lagrangian function
• Weak and strong duality
• Second-order conditions