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.

Students who have a valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the teacher and with the Department of Architecture and Design's disability referent (https://architettura.unige.it/commissioni_e_referenti_dipartimento).

SYLLABUS/CONTENT

The course contains elements of Mathematical Analysis and Geometry.

Algebra and Geometry 

Sets: union, intersection, complement, functions, domain, codomain, image, composition, invertible functions, right and left inverses. Injective, surjective and bijective functions.

Matrices: Operations with matrices and their properties, Gaussian form, reduced Gaussian form, rank, determinant, inverse matrix, completion of low-rank matrices.

Linear systems: reduction to row echelon form of linear systems using Gaussian elimination method, existence and multiplicity theorem for solutions of linear systems. Homogeneous systems, positive solutions of linear systems.

Complex numbers: Algebraic representation, geometric representation, modulus, conjugate, inverse. Trigonometric representation and polar coordinates. Complex exponential. Solving equations.

Vectors: Geometric vectors. The vector spaces R² and R³ and their properties. Bases and dimensions of vector subspaces of R² and R³.

Elements of geometry in the plane and in space: Lines, planes, conics. Cartesian form, parametric form, point-to-line and point-to-plane distance. Pencils of lines and pencils of planes. Parallel, skew and intersecting lines in 3-dimensional space.

ANALYSIS Real functions of one real variable: Basic concepts and elementary functions.

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

Derivatives and their applications: Definition and geometric meaning. Differentiation rules. Graph of the derivative. Fermat's theorem. Convexity and concavity. Function analysis.

Integral calculus: Area and estimation using finite sums: definite integral. Integrable functions and integrability of continuous functions. Fundamental theorem of integral calculus. Indefinite integral. Integration techniques and integrals of elementary functions. Examples of double integrals.

Ordinary differential equations: General integral and Cauchy problem. Separable variable equations. Second-order homogeneous and non-homogeneous equations with constant coefficients.

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)

C. Marcelli, Analisi Matematica 1. Esercizi con richiami di teoria., Pearson, 2019

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

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

In agreement with the Academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

• You must register at least one week in advance at https://servizionline.unige.it/studenti/esami/prenotazione. For organizational reasons, registrations will not be accepted after the booking deadline.

  The exam consists of a written test and an oral test.
  The written test is composed of two parts: one covering algebra and geometry, and the other covering analysis.

  The written test includes:

• A section on algebra and geometry (3 exercises to be solved in 90 minutes)

• A section on analysis (3 exercises to be solved in 90 minutes)

• The written test is considered passed if the student obtains at least 16 points out of 30 in each section, with anoverall average of 18 or higher.
  For example:

• A score of 16/30 in algebra and geometry and 20/30 in analysis is a passing grade.

• A score of 18/30 in algebra and geometry and 17/30 in analysis is not a passing grade.

• The two parts of the written test may be passed in different exam sessions.

  Students may take the oral exam only after passing the written test.
  The oral exam is conducted upon request by either the student or the instructors, and it covers the entire course syllabus.
  The final grade after the oral exam may be higher or lower than the written exam grade, and failing the oral exam may result in not passing the course.

   

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.

FURTHER INFORMATION

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.