## MATHEMATICS I

Elementary properties of integers and rationals

Symbolic algebra

Coordinates on the line

Coordinates in the plane

Trigonometric functions (these will be reviewed

anyway)

Written and oral exam, both compulsory. The written exam consists of problems. Midterm written exams (two) can be taken instead of final written exam.

The course focuses on basic techniques (rather

than formal proofs) in Linear Algebra, Vector Calculus and Real Variable Calculus. Students will get a grasp of

basic concepts in Analysis, such as limits, continuity,

differentiability, local and asymptotic behaviour of elementary functions.

Linear systems and eigenvector theory are introduced from a geometric viewpoint.

Complex numbers (if time allows) will be introduced

in both the algebraic form a+ib (to highlight similarity with vector sum), but

also in polar form, leading to simplification in

derivation and integration of

trigonometric functions.

Note that Integration will be developed

further in the Course "Matematica II".

Real powers, absolute value, roots.

Coordinates. Cartesian and

parametric equations. Sums of vectors.

Scalar and vector product.

Linear systems. Matrices and determinants. Independence, bases. Projections. Matrix of an operator. Eigenvectors.

Complex numbers. Euler Formula. Roots.

Functions and graphs. Invertible functions.

Convergent sequences and series. Binomial theorem.

Theorems about limits. Properties of continuous functions.

Differentiation. Mean Value Theorem. If the sign of f' is constant

in an interval, then f is monotonic in that interval.

f'' and convexity. Criteria for maxima and minima in stationary points.

Taylor expansions.

Definite and indefinite integrals. Fundamental Theorem of Calculus.

Integration by change of variables and by parts.

Bramanti--Pagani--Salsa: Matematica. Calcolo infinitesimale e algebra

lineare

(ZANICHELLI)

OR

Bramanti--Pagani--Salsa: Analisi matematica 1 con elementi di algebra lineare

(ZANICHELLI)

In presence with Problem classes (32 hours).

1) Readiness to offer individual assistance also in English and French language to incoming students

2) Availability of supporting material and bibliographic references also in english language.

3) Readiness to accept examination of incoming students also in a foreign language (English-French)