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MATHEMATICS

MATHEMATICS II

Program

EN IT

Updated A.Y. 2020-2021

Calculus

- Integral Calculus. Definite and indefinite integrals, Integral properties, The fundamental theorem of calculus, Integration by parts and integration by substitution, Improper integrals.

Linear Algebra

- Linear Spaces. The algebra of vectors, Euclidean Spaces, Inner product, Linear independence.

- Matrices. Matrix algebra, Determinant, Inverse Matrix. 

- Systems of Linear equations. The Gauss Elimination Algorithm, Rouche Capelli theorem.

- Eigenvalues and eigenvectors, Diagonalization.

Optimization

- Calculus of several variables. Domain, Partial derivatives, gradient, hessian matrix. Stationary points. Countour curves.

- Unconstrained optimization. First and second order conditions

- Constrained optimization with Equality constraints. First and second order conditions

- Optimization on a set

 

Suggested Books:

Knut Sydsaeter, Peter Hammond and Arne Strom, Essential Mathematics for Economic Analysis

Carl P. Simon, Lawrence Blume, Mathematics for Economists

Teaching Material

Available on the webpage of the course and on Teams:

  • Slides of the course
  • Additional Exercises

Available on Moodle: 

  • Self-check tests are uploaded on Friday every week

 

The final Syllabus will be available only at the end of the course. 

You can find below a detailed program which will be update on a daily basis during the teaching period.

 

Detailed Program

22/02/2021 Antiderivative. Theorem: if a function has one antiderivative, then it has infinitely many antiderivatives (with proof). Indefinite integrals: definition. Integrals of Elementary Functions. 

23/02/2020 Integrals of Elementary Functions., Integration by substitution, integration by parts, Approximation of an area by Riemann sums

24/02/2020 Definite integral, Properties of definite integrals, The Fundamental Theorem of Calculus: statement and applications, The Mean Value Theorem: statement and geometric interpretation. 

01/03/2020 Improper integral. Matrix, Operations with matrices: sum and multiplication by a scalar (definition and properties). The null matrix. 

02/03/2020 The transpose of a matrix, Vectors, operations with vectors: sum, multiplication by a scalar, inner product (definition and properties). Linear combination of vectors. The standard vectors in R^n: definition and properties (with proof). Matrix-vector multiplication (definition and properties). The identity matrix: definition and property (with proof).

03/03/2020 Matrix product (definition and properties), Matrix product is not commutative (two counterexamples), Diagonal, Lower Triangular and Upper Triangular matrices.  Inverse of a matrix (definition), The inverse of a matrix is unique (with proof), Inverse of a 2x2 matrix (with proof), necessary and sufficient condition for invertibility, The determinant of a matrix: cofactor expansion by rows and by columns.