# MATHEMATICS

MATHEMATICAL ANALYSIS 1 (6 credits) | GEOMETRY AND ALGEBRA (4 credits) |

## Learning outcomes of the course unit

MATHEMATICAL ANALYSIS 1:

The course will introduce the basic concepts of Mathematical Analysis. Besides providing computing tools, classes will be aimed at stimulating a deeper understanding of the ideas and methods of mathematical thinking. The main goals are: 1) improving the skills of the student in learning new mathematical tools; 2) improving the ability of the students to apply the newly learned theoretical notions in the solution of problems and exercises;

3) improving to student's skill in communicating with clarity and precision the mathematical concepts acquired during the course.

GEOMETRY and ALGEBRA:

Knowledge and understanding:

basic Linear Algebra and Geometry of the space.

Applying knowledge and understanding:

a) solve systems of linear equations;

b) diagonalize matrices;

c) solve easy problems of analytic geometry;

d) recognize the type of a conic and write its canonical form.

Communication and learning skills:

properly express themselves with mathematical language.

## Prerequisites

Matematica preuniversitaria di base

## Course contents summary

The course is made up of two parallel sub-courses: one sub-course, corresponding to 6 CFU, is named Mathematical Analysis 1 and is held by Prof. Massimiliano Morini, the other one, held by Prof. Lucia Alessandrini, is named Geometry and Algebra and is worth 4 CFU.

The course Mathematical Analysis 1 serves as an introduction to the main concepts of the analysis of functions of one real variable: continuity, limits, derivatives, and integrals.

The first part of Geometry and Algebra is devoted to Euclidean Geometry in the space (vectors, lines, planes), while the second part of the course is devoted to matrices and linear systems. In the third part we study vector subspaces of Rn, linear maps and the diagonalization of linear operators.

## Course contents

MATHEMATICAL ANALYSIS 1:

- Basic concepts of set theory

- Rational numbers

- Tha axiom of continuity and the properties of real numbers

- inf and sup of a set of real numbers

- Functions: definitions and examples.

- Limits of functions.

- Properties of the limits.

- Continuous functions and their application to the computation of limits.

- The fundamental properties of continuous functions: Weiestrass Theorem and the Intermediate Value Theorem.

- The concept of derivative: heuristics, geometric motivation and rigorous definition.

- Examples of non-differentiable functions.

- Differentiability implies continuity.

- Rules for the computation of derivatives.

- Search of minimum and maximum points of a differentiable function: Fermat's Theorem.

- Rolle's and Lagranges's Theorems and their important consequences on the study of the monotonicity properties of functions.

- -Convexity/concavity and second derivatives.

- Qualitative study of functions.

- The notion of antiderivative.

-Main rules for the computations of antiderivatives: change of variables and integration by parts.

- The notion of integral

- The Fundamental Theorem of Integral Calculus.

-Examples and applications.

GEOMETRY and ALGEBRA:

Euclidean Geometry in the space.

1. Vectors and its operations. Coordinates. Scalar product. Distances and angles. Vector product in R3.

2. Three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position between two lines in the space; skew lines. Equation of a plane. Quadric surfaces.

Vectors, matrices, linear systems.

3. The n-dimensional space Rn and its properties.

4. Matrices and their properties. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.

5. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.

6. Linear subspaces of Rn. Linear combinations of vectors: linear dipendence/indipendence. Generators, bases and dimension of a vector subspace.

Linear maps.

7. Linear maps. Definition of kernel and image. Matrix representation of a linear map. Isomorphisms and inverse matrix.

8. Eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable operators.

## Recommended readings

MATHEMATICAL ANALYSIS 1:

1) E. Acerbi, G. Buttazzo: "Matematica preuniversitaria di base". Pitagora Editrice, Bologna.

2) E. Acerbi, G. Buttazzo: "Analisi Matematica ABC". Pitagora Editrice, Bologna.

3) A. Guerraggio: "Matematica per le Scienze". Editore: Pearson

GEOMETRY and ALGEBRA:

1) ALESSANDRINI, L., NICOLODI, L., GEOMETRIA A, ED. UNINOVA (PR) 2004.

## Teaching methods

MATHEMATICAL ANALYSIS 1:

Traditional lectures held by the teacher and homework assignments aiming at stimulating "active" learning of the concepts taught in class.

GEOMETRY and ALGEBRA:

In the lectures we shall propose formal definitions and proofs, with significant examples and applications, and several exercises. Exercises are an essential tool in Linear Algebra; they will be proposed also in addition to lectures, in a guided manner.

## Assessment methods and criteria

MATHEMATICAL ANALYSIS 1:

Written exam

GEOMETRY and ALGEBRA:

Learning is checked by a written exam. The student can also perform two written exams during the course, to avoid the final written exam.

In the written exam the student must exhibit basic knowledge related to Linear Algebra and Euclidean Geometry in the space.