The course’s main objective is to convey, in a unified way, the learning of the main basic mathematical tools and of its language. It is directed at learners coming from diversified educational backgrounds. More specifically, the syllabus will deal with topics preparatory and in support of the parallel course of architectural drawing, and of the subsequent courses of building physics, real estate evaluation, as well as of the structural matters. Wherever possible, links and synergies with such disciplines, typical of the course of Architecture, will be outlined. For each of the arguments touched upon, the basic notions and the principal methods of analysis and resolution will be presented. The course is organized with the objective to convey to the student the capacity to identify, interpret, apply and present results and problems of a mathematical character, aiming at the logic of the solution more than at the complexity of calculus, which can at a later stage be resolved with the help of appropriate software machinery

The students will acquire knowledge and skills below described, tied to the essential mathematical tools, specifically in order to be able to: - understand - classify - resolve differential calculus problems in one variable, integral calculus problems, linear algebra and geometry. As well as: - identify - examine - reorganize the mathematical concepts and objects used within the disciplinary areas provided for by the educational path. These include matrices, vectors, plane’s geometrical entities, surface area calculus, functional relations and their classifications. Special attention will be devoted to spatial visualizations and to their mathematical description, thus the student shall prove able to: - recognize from its graphic representation the properties of a function, as well as: - represent in a graphic form elementary geometrical entities and functions of a single real variable. The course will teach the students how to analyze, formalize through a mathematical model and solve simple problems of varying nature that they will encounter in their profession, while appraising the most appropriate analytical and geometrical instruments.

Knowledge of mathematical concepts typical of completed high school/entry stage of college is required. More specifically, the student shall know how to solve first and second degree equations and inequalities, fractional equations and inequalities, systems of inequalities, products and divisions of polynomials; he/she shall know basic concepts in goniometry and trigonometry, logarithms and exponentials; he/she shall know the main properties of powers, of the absolute value, of the plane’s Euclidean geometrical entities.

Linear algebra and geometry. Matrices and their operations: addition, product with a scalar, matrix multiplication, transposition; submatrices; formulas to calculate the determinants of square matrices; rank of a matrix. Vectors and their operations: geometric representation and representation with Cartesian orthonormal components; magnitude an direction; addition, subtraction and linear combination of vectors; scalar, cross and scalar triple products. Linear systems: Cramer’s rule and Rouché-Capelli theorem; linear algebra applications for the study of planes and lines; basics of plane geometry. Differential calculus. Functions: domain, codomain and graphic representation; main symmetries; elementary functions (rational and irrational functions, exponential and logarithmic functions, trigonometric and hyperbolic functions); elementary operations between functions, reflections and translations; limits of functions and their determination in straightforward cases; indeterminate forms and examples of special limits; continuity and related theorems; derivability, computation of derivatives and related applications, also in the resolution of calculus problems; first derivative and tangent line; main theorems concerning monotonicity and concavity/convexity; behavior at the boundary of the domain; qualitative graph of a function. Single variable integral calculus. Single variable integral calculus: main properties; the definite integral and the surface area calculus; the indefinite integral and the search for simple primitives.

The course is structured in 50 hours of frontal class (theoretical) lectures and 30 hours of lectures devoted to exercises, that is a subdivision of approximately 60% and 40% of those activities. The first part of the course will be mainly devoted to the subjects of linear algebra and geometry. The second part will be mainly devoted to the subjects of differential and integral calculus. To both parts a considerable portion of time of the course will be dedicated, although not necessarily in exactly equal proportions.

R. A. Adams, Calculus, a complete course, Pearson Canada S. Lipschutz, M. Lipson, Linear Algebra, Schaum’s outlines The teacher will make notes and/or exercises available online, on the students-teachers platform. Materials for further readings will be communicated during class, if needed.

Modalità di esame: Prova orale facoltativa; Prova scritta su carta con videosorveglianza dei docenti;

The exam aims at ascertaining the acquisition of knowledge that allows one to comprehend and classify single-variable differential calculus problems, integral calculus, linear algebra and geometry, such as detailed in the syllabus; to identify matrices, vectors, the plane’s and space’s geometrical entities, surface area calculus, functional relations and their classifications. At the same time, the exam will ascertain the acquisition of the corresponding skills, such as knowing how to solve single-variable differential calculus problems, integral calculus, linear algebra and geometry. Thus, to know how to reorganize the concepts of matrix, vector, geometrical entity in the plane and in space, surface area, function. The ability to recognize the graph of a function or the skill to represent it will be verified. The written exam consists of a series of 8 open-ended exercises, of which 2 with a maximum of 7 points available and 6 with a maximum of 3 points available. Maximum points shall be awarded to an exercise if it is complete, correct, logically sound and clearly presented. Any unanswered exercise is valued zero. The written exam has a duration of 2 hours, subdivided into two parts of 1 hour + 1 hour, with handing of 4 exercises after each hour. Each part consists of three exercises of maximum 3 points and one of maximum 7 points. In case of impossibility to take the written exam in physical presence, it will consist of “prova scritta su carta e videosorveglianza”, meaning “written test with video surveillance” (with the possibility of being recorded). In such case, technical instructions on the procedure for carrying out and handing (uploading) the paperwork will be given during the course. It will be necessary to have at least one device with an online camera available (e.g. PC or smartphone). It may be necessary to have a second device available. The written exam is passed with a mark of at least 18/30. Use of personal notes, textbooks, printed notes and calculator is not permitted. It is possible to use an inventory of formulas specifically prepared by the teacher, which will be rendered available online through the portal. The final grading is composed of the written exam’s mark and, upon explicit request by the student or as an option exercised by the teacher, by the mark of an additional oral examination, which can make the final grading vary in positive as well as in negative. The oral examination must be taken during the same exam period during which the written exam was passed. The oral examination is not administered following a written exam which was not passed.

Modalità di esame: Prova scritta (in aula); Prova orale facoltativa; Prova scritta su carta con videosorveglianza dei docenti;

The exam aims at ascertaining the acquisition of knowledge that allows one to comprehend and classify single-variable differential calculus problems, integral calculus, linear algebra and geometry, such as detailed in the syllabus; to identify matrices, vectors, the plane’s and space’s geometrical entities, surface area calculus, functional relations and their classifications. At the same time, the exam will ascertain the acquisition of the corresponding skills, such as knowing how to solve single-variable differential calculus problems, integral calculus, linear algebra and geometry. Thus, to know how to reorganize the concepts of matrix, vector, geometrical entity in the plane and in space, surface area, function. The ability to recognize the graph of a function or the skill to represent it will be verified. The written exam consists of a series of 8 open-ended exercises, of which 2 with a maximum of 7 points available and 6 with a maximum of 3 points available. Maximum points shall be awarded to an exercise if it is complete, correct, logically sound and clearly presented. Any unanswered exercise is valued zero. The written exam has a duration of 2 hours, subdivided into two parts of 1 hour + 1 hour, with handing of 4 exercises after each hour. Each part consists of three exercises of maximum 3 points and one of maximum 7 points. In case of impossibility to take the written exam in physical presence, it will consist of “prova scritta su carta e videosorveglianza”, meaning “written test with video surveillance” (with the possibility of being recorded). In such case, technical instructions on the procedure for carrying out and handing (uploading) the paperwork will be given during the course. It will be necessary to have at least one device with an online camera available (e.g. PC or smartphone). It may be necessary to have a second device available. The written exam is passed with a mark of at least 18/30. Use of personal notes, textbooks, printed notes and calculator is not permitted. It is possible to use an inventory of formulas specifically prepared by the teacher, which will be rendered available online through the portal. The final grading is composed of the written exam’s mark and, upon explicit request by the student or as an option exercised by the teacher, by the mark of an additional oral examination, which can make the final grading vary in positive as well as in negative. The oral examination must be taken during the same exam period during which the written exam was passed. The oral examination is not administered following a written exam which was not passed.