20355 - MATHEMATICS - PREPARATORY COURSE
Course taught in English
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 the geometrical aspects and focusing on the economic interpretation.
- Euclidean spaces: geometric and algebraic approaches. Vectors in R^n. Operations with vectors. Matrices. Linear Space: linear dependance and independance. Dimension and bases of the linear space. Examples. Straight lines and planes in R^3. Linear systems: structure of solutions. Linear functions between euclidean spaces. Representation theorem. Eigenvalues and eigenvectors of a linear transformation. Spectral theorem for symmetric matrices.
- Definitions and applications. Examples.
Curves in the plane and in the 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. Differential. Higher order derivatives. Derivative of a composite function. Hessian matrix. Implicit functions. Implicit function theorem. Jacobian matrix.
- Unconstrained optimization. The first order sufficient conditions. Fermat's theorem. Taylor polynomial of order two. Concavity and convexity. Second order sufficient conditions. Local-global theorem. Constrained oprimization. Lagrange multipliers technique. Meaning of multipliers.