20355 - PRECORSO DI MATEMATICA / MATHEMATICS - PREPARATORY COURSE

Deep familiarity with some topics generally carried out in the first-level courses in Mathematics is essential for a good understanding of the contents of the Advanced Mathematics for Economics and Social Sciences Course. These arguments are practically and theoretically reviewed along the Preparatory Course, combining the analytical approach to geometrical aspects and focusing on the economic interpretations.

Linear Algebra:

• Euclidean spaces: geometric and algebraic approaches. Vectors in R^n. Operations with vectors. Matrices. Linear space: linear dependence and linear independence. Dimension and bases of a linear space. Examples. Straight lines and planes in R3. Linear systems: structure of solutions. Linear functions between euclidean spaces. Representation theorem. Eigenvalues and eigenvectors of a linear transformation. Spectral theorem for symmetric matrices.

Quadratic forms:

• Definitions and applications. Examples.

Curves in the plane and in space:

• Straight lines in space. Parametric representation of a trajectory. Speed and tangent vector.

Functions in several variables.

• Level lines and contour map. Partial derivatives, gradient. Tangent plane. Differential.Higher order derivatives. Derivative of a composite function. Hessian matrix. Implicit functions. Implicit function theorem. Jacobian matrix.

Optimization problems:

• Unconstrained optimization. The first-order sufficient conditions. Fermat's theorem. Taylor polynomial of order 2. Concavity and convexity. Second order sufficient conditions. Local-global theorem. Constrained optimization. Lagrange multipliers technique. Meaning of multipliers.

S. CERREIA, M. MARINACCI, E. VIGNA, Principles of Mathematics for Economics, draft version, 2016.