# 65926 - Mathematical Institutions 1

### Course Unit Page

• Teacher Silvia Foschi

• Credits 8

• SSD MAT/05

• Language Italian

• Campus of Ravenna

• Degree Programme First cycle degree programme (L) in Environmental Sciences (cod. 8011)

• Course Timetable from Sep 27, 2022 to Dec 21, 2022

## Learning outcomes

Upon completion of this course, the student knows the basic tools of mathematical analysis and linear algebra. He/she is able to use mathematical tools for the study of other disciplines.

## Course contents

Analysis. Sets. Relations. Maximum, minimum, lower and upper extremes of a set. Functions. Even, odd, periodic, monotonic (increasing, decreasing), injective, suriective functions. Composition of functions. Invertible functions and their inverse. Fundamental functions.

Real functions of a real variable: limits and their theorems, calculation of limits. Continuity of a function and its theorems: Bolzano's theorem on intermediate values, zeros theorem, Weierstass theorem. Discontinuity. Asymptotes.

Incremental ratio. Derivative. Rules for the calculation of derivatives. Points of non-derivability. Continuity and derivability. Fermat's theorem. Rolle's theorem. Lagrange's intermediate value theorem. Monotony test. Search of relative and absolute extremes. Theorem of de l'Hospital. Higher order derivatives. Taylor polynomials and local approximation of functions. Concavity and convexity, bending, study of a function and its graph.

Integral according to Riemann: integrability and integral. Fundamental theorems of integral calculus. Integration of elementary functions. Integration of fraternal algebraic functions, method of simple fractions. Integration by parts. Integration by substitution (or with change of variable).

Complex numbers: algebraic, Cartesian, trigonometric, exponential representation. Operations in C. N-hex roots of complex numbers. Solving equations in C, geometric places.

Linear Algebra. Matrices and their operations. Elementary operations on the rows of a matrix. Gauss reduction method for the rank of a matrix and for the resolution of linear systems. Homogeneous linear systems. Determinant of a matrix. Inverse matrix, Gauss-Jordan method.

Vectors. Vector spaces. Linear dependence between vectors. Generators of a vector space. Base of a vector space. Scalar product. Orthogonality between vectors. Orthogonal basis. Vector product. Eigenvectors. Eigenspaces.

Analytic geometry in space. Equation of a plane, line equation in parametric and Cartesian form. Orthogonality and parallelism between planes and lines. Point to line distance. Point-plane distance.

Daniele Ritelli. Lectures in Mathematical Analysis 3rd Edition. Esculapio 2019.

ISBN: 9788874888870

Daniele Ritelli, Massimo Bergamini, Anna Trifone, Fundamentals of Mathematics, Zanichelli

M. Barnabei,F. Bonetti, Linear systems and matrices, Pitagora Editrice, Bologna

## Teaching methods

Classroom lectures with use of video projector. Assignment of work to be done independently. Availability of notes with exercises.

## Assessment methods

Written test. Possible further optional oral test.

## Teaching tools

Combined use of blackboard and video projector.

The didactic material presented in class will be made available to the student in electronic format through the institutional portal of the University. Username and password are reserved to students enrolled in the Degree Course in Environmental Sciences, Ravenna Campus, University of Bologna.

The teacher will reply to e-mail messages, duly signed by the student with Name, Surname and matriculation number, concerning appointment requests or topics that are not covered by the course information presented here.

## Office hours

See the website of Silvia Foschi