# MATHEMATICS 1

OVERVIEW

This course equips the students with the mathematical principles and the tools needed to study structural disciplines and design, and to understand architectural morphology, and physical, technological, economical, social and urban models.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course aims is to provide the students with the mathematical tools which are needed to tackle any problem with a scientific approach.

AIMS AND LEARNING OUTCOMES

The course aims to provide the basic tools that allow the students to tackle any topic with a scientific approach and to stimulate the three-dimensional and aesthetic sense needed to an architect. More specifically, the aim of the course is to provide the mathematical principles and tools necessary to tackle the study and understanding of structural and design disciplines, of physical, technological, economic, social and urban planning models.

At the end of the course, students will be able to: solve linear systems, operate on vectors, recognize planes and lines in 3D, master the fundamental concepts of differential and integral calculus for one variable functions, qualitatively study the graphs of functions, solve simple differential equations, and work with complex numbers. Furthermore, we expect the ability to state and demonstrate some basic theorems of mathematical analysis.

At the end of the course we expect a critical understanding of the subject, the ability to distinguish different situations on specific examples and to make reasoned choices, justifying the chosen procedures. Some ability in the computations and a well-argued exposition of the theory is also expected.

PREREQUISITES

A good knowledge of the mathematical topics covered in secondary school is needed. In particular, we assume well understanding of polynomials, equations, inequalities, trigonometry, Euclidean geometry (areas and volumes of elementary geometric figures), elements of analytical geometry.

Teaching methods

Lectures and exercises on the blackboard. A tutor is available for further explanations and exercises; exercises are provided for students' autonomous work.

SYLLABUS/CONTENT

The course contains topics of Mathematical Analysis and Geometry.

**GEOMETRY**

**Linear systems:** solution of linear systems using the Gauss elimination method, existence theorem and multiplicity of solutions of linear systems

**Matrices:** Operations with matrices, rank, determinant, inverse matrix.

**Vectors:** Geometric vectors. The linear spaces R^2 and R^3 and their properties. Bases and dimension of vector subspaces of R^2 and R^3.

Elements of geometry in the plane and in space. Lines, planes, conics and quadrics.

**ANALYSIS**

**Real functions of one real variable**. Basic notions and elementary functions.

**Limits and continuity**. Definition, calculation of limits, fundamental theorems.

**Derivatives and their applications.** Definition and geometric meaning. Derivation calculus. Graph of the derivative. Fermat's theorem. Convexity and concavity. Qualitative study of the graph.

**Integrals.** Area and estimate by finite sums: definite integral. Integratable functions and integration of continuous functions. Fundamental theorem of integral calculus. Indefinite integral. Integration and integral techniques of elementary functions.

**Complex numbers.** Algebraic representation, modulus and conjugate. Trigonometric representation and polar coordinates. Complex exponential. Solving equations.

**Ordinary differential equations:** General integral and Cauchy problem. Equations with separable variables. Equations with constant coefficients of the second order homogeneous and not homogeneous.

RECOMMENDED READING/BIBLIOGRAPHY

*M. Abate, C. de Fabritiis*, **Geometria analitica con elementi di algebra lineare.** McGraw-Hill Libri Italia, 2006

*J. Hass, M.D. Weir, G.B. Thomas, ***Analisi Matematica 1,** Pearson, 2018 (English edition available)

*G. Crasta, A. Malusa*, **Elementi di Analisi Matematica e Geometria con prerequisiti ed esercizi svolti**, La Dotta, 2015

## TEACHERS AND EXAM BOARD

**Ricevimento:** Friday 14-16

**Ricevimento:** By appointment wich can be fixed in person or via email : villa@dima.unige.it

Exam Board

SILVIA VILLA (President)

ERNESTO DE VITO

SANDRO BETTIN

## LESSONS

Teaching methods

Lectures and exercises on the blackboard. A tutor is available for further explanations and exercises; exercises are provided for students' autonomous work.

LESSONS START

In agreement with the Academic calendar.

## EXAMS

Exam description

The exam consists of two intermediate written tests or a final written test. The written tests contain exercises and a theoretical part (both to be performed without books or notes). The exam is passed if a grade higher or equal to 18/30 is awarded for each intermediate test, or if the written exam is accompanied by a mark higher than or equal to 18/30.

Failure to pass the two intermediate tests or a final test oblige the student to participate to a subsequent round. The exam can be completed by an oral test.

The first intermediate test will be held during the lessons break, the second one at the end of the course. The student can participate to the second test only if he has successfully passed the first ont. Not successfully completed tests will be discarded.

**Intermediate written tests**

The tests consist of a theoretical question, to be answered in 15 minutes and three exercises, to be carried out in an hour and thirty minutes. It is necessary to present the complete development of the exercises, properly justifying the provided answers.

**Written tests at a round**

The tests consist of two theoretical questions, to be answered in 20 minutes, and five exercises, to be solved in two hours.

It is necessary to deliver the complete development of the exercises, properly justifying the answers provided and the steps taken.

**Oral exam**

One can access to the oral part only after passing the written test. The oral examination can be done at the request of the student or the teacher and covers the entire course's program. The grade obtained at the end of the oral examination may be higher or lower than the one obtained in the written test, and may eventually lead to failure to pass the exam.

**Exercises and theoretical questions**

The exercises that will be proposed in the written tests concern the entire program of the course, available on Aulaweb. The theoretical questions will be related to definitions, examples, general presentation of a topic, statements of theorems reported in the cited program with or without an asterisk. Demonstrations may also be requested, including those that appear in the program accompanied by an asterisk.

**Evaluation methods for the intermediate written or appeal tests**

Exercises: the maximum score that can be obtained is 27.

Theoretical part: the maximum score that can be obtained is 5.

If the student passes an intermediate test, the grade of the test is given by the sum of the scores of theexercises and the theory. If the student passes both intermediate tests, the exam grade is given by the average of the marks of the two intermediate tests.

If the student passes a round, the exam grade is given by the sum of the scores of the part of exercises and of the theory.

A bonus up to 2 points is given for the participation to tutoring activities.

Assessment methods

Evaluation of the written test and of the possible oral exam. The educattional goal is achieved to the extent that the student is capable of solving exercises of similar difficulty to those solved during classes and has a critical knowledge of the fundamental contents of the course.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

08/06/2021 | 09:00 | GENOVA | Compitino | |

08/06/2021 | 09:00 | GENOVA | Scritto | |

06/07/2021 | 09:00 | GENOVA | Scritto | |

06/07/2021 | 09:00 | GENOVA | Scritto | |

14/09/2021 | 09:00 | GENOVA | Scritto |