The course provides the principles and tools of mathematics needed to address the study and understanding of structural and design disciplines, architectural morphology and physical, technological, economic, social and urban models. The course includes elements of analytical geometry in the plane and in space, linear and vector algebra with its applications, differential and integral calculus, mathematical models. Moreover, the course includes essential tools for the study of structural disciplines, technical physics and also architectural geometry, in view of the new methodologies oriented to parametric design. Finally, the course also pursues the goal of providing the student with the basic tools of mathematical logic that are crucial for the education of an architect.

An innovative teaching activity (1 cfu) is planned by means of a MOOC (Massive Open Online Course) expressly conceived for this course. It will support the students in their preparation jointly with specific tutoring actions.

Risultati di apprendimento attesi

After passing the exam, the student:

-understands the concepts of: (a) function of a real variable together with its graphic representation; (b) limit and continuity;

-applies the knowledge and understanding of differential calculus to provide the qualitative representation of the graph of elementary functions;

-applies the knowledge and understanding of the essential notions and main rules of integral calculus, with particular emphasis to the calculation of areas and centers of gravity;

-is able to solve, by resorting to Gauss elimination algorithm, the systems of linear equations in n unknowns, also dependent on a parameter and to understand the theoretical results determining the existence and multiplicity of solutions together with the concepts related to the dependence or independence of the equations themselves;

-understands the use of reference and coordinate systems in three-dimensional space and is able to describe systems of lines and planes, both in parametric and Cartesian form.

-is also able to understand and apply the conditions of parallelism/intersections between planes and he is able to compute orthogonal projections;

-knows and understands the concepts of: vector, sum and multiple of vectors, components, scalar and vector product, mixed product together with their applications and geometric interpretations. He is also able to apply the knowledge and understanding of basic tools of vectorial calculus to compute resultants and moments of force systems and make judgement of their simplest properties.

-is also able to express the correct mathematical formulation of problems stemming from typical applications of the course of study in Architectural Design.

-is able to understand texts and references containing mathematical concepts of interest for Architecture. If required by other courses in Architectural Design, he is also able to autonomously acquire further and adequate mathematical skills.

Argomenti trattati

-Numerical sets. Basic elements of elementary algebra and trigonometry. Elements of analytical geometry. Cartesian plane and graph of a function. Elementary and trigonometric functions. -Matrices, linear transformations and matrix operations. Determinant and rank. Vectors of R ^ n. Linear combinations and linear independence. Theory of systems of linear equations: existence and multiplicity of solutions. Homogeneous systems. -Vector algebra in three-dimensional space. Scalar product, vector product, mixed product. Applied vectors and forces. Resultant and moment of a system of forces. -Geometry of space: points, lines and planes. Parametric description and Cartesian description of lines and planes. Parallelism, intersections, orthogonal projections. Overview of curves and parametric surfaces. -Mathematical models and functions of a real variable. Limit and continuity. Differential calculus: fundamental theorems, derivative and differential, approximation formulas, qualitative study of functions. Applications to optimization problems. -Definite integral. Theorem of the mean and fundamental theorem of the integral calculus. Primitives and elementary rules of integration. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- One or more of the following topics may OPTIONALLY be carried out: -Elemental differential equations. Second order linear differential equations with constant coefficients. -Functions of two real variables: initial definitions, level curves, partial derivatives, directional derivative. -Parametric description of curves and surfaces and their elementary geometrical properties. -Equivalent force systems, torques, center of gravity, straight line of application of the resultant. -Principal directions and eigenvalues for linear transformations of the plane and space. -Method of least squares for rectangular linear systems.

Prerequisiti

A knowledge of elementary mathematics is required: Cartesian plane; equation of the line, of the parabola and of the circumference; first and second order equations in one variable, first and second order inequalities in one variable. Elementary Euclidean Geometry of plane and space. Polynomials in one variable. Basics of elementary trigonometry. Elementary functions: logarithm, exponential, sine, cosine and tangent.

Modalità di valutazione

The assessment of the students is organized according to the schedule provided by the School's Academic Calendar.

The assessment is based on written tests containing a number and type of exercises adequate to verify: (a) the achievement of a sufficient level of knowledge and understanding of the contents of the course; (b) the ability to apply the knowledge to given problems.

The written tests, at the discretion of the teacher, can also include one or more theoretical questions aimed at verifying the knowledge and understanding of the main theoretical concepts introduced during the course.

After each written test, the teacher, in his complete discretion, can complement the assessment procedure with an oral examination.

Approximately halfway through the teaching period there is a mid-term written test assessing the material of the course covered until that moment.

At the end of the semester a second written test is scheduled. All students are admitted to the second test, regardless of the outcome of the first one.

The partial results achieved independently in the two tests maintain their validity until the end of the summer session (end of September).

The overall evaluation of the student is obtained by taking the average of the outcomes of the first (mid-term) and second test. If both the first and second test have a grade larger or equal than 14/30 and the average of the two grades is greater or equal than 18/30 then the student passes the exam.

If either the first or the second test has a grade less than 14/30 or the averge is less than 18/30, the student has to take the written test during one of the subsequent dates according to the schedule provided by the School's Academic Calendar. At his discretion, the student can opt to exploit the validity of one of the two partial results. In this latter case a suitable and consistent version of the written test will be provided.

In case the student opt for exploiting the validity of one of the two partial results, the overall evaluation is obtained by taking the average between the selected partial result and the actual one (only if this latter is larger or equal than 14/30). The student passes the exam if the average is greater or equal than 18/30.

If the student does not opt for exploiting the validity of partial results, the exam is considered to be passed if a grade larger or equal than 18/30 is obtained.

Bibliografia

F. Caliò. A. Lazzari, Elements of Mathematics, Editore: Esculapio, Anno edizione: 2018
J. Stewart, Calculus, Editore: Cengage Learning Brooks/Cole

Software utilizzato

Nessun software richiesto

Forme didattiche

Tipo Forma Didattica

Ore di attività svolte in aula

(hh:mm)

Ore di studio autonome

(hh:mm)

Lezione

44:00

66:00

Esercitazione

36:00

54:00

Laboratorio Informatico

0:00

0:00

Laboratorio Sperimentale

0:00

0:00

Laboratorio Di Progetto

0:00

0:00

Totale

80:00

120:00

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