Building engineering and architecture

Syllabus for the Academic Year 2025/2026 n-Dimensional Euclidean Spaces Properties of the space ℝⁿ and definitions of distance (metric), norm, and inner product. Elements of topology in ℝⁿ: neighborhoods, open and closed sets, accumulation points, isolated points, boundary points, and closure of a set. Bounded and compact sets, connected open sets. Review of Euclidean and analytic geometry in the plane and in space: parallelism and orthogonality among vectors, lines, and planes; Cartesian equations of cylinders, spheres, ellipsoids, cones, and paraboloids. Parametric Curves Physical motivation and definition of a curve in ℝⁿ. Simple, closed, C¹, regular, and piecewise regular curves. Unit tangent vector and tangent line to a curve at a point. Orientation of a curve, admissible reparametrizations, and equivalent curves. Length of a curve and rectifiability of C¹ curves. Significant examples of curves. Introduction to Real-Valued Functions of n Real Variables Domain and graph of a function. Limits of functions: definitions, properties, and techniques for evaluating limits in indeterminate forms, using the necessary condition for the existence of a limit via path testing, and the sufficient condition via bounding arguments. Continuous functions: definition and properties. Absolute extrema and the Weierstrass theorem. Differential Calculus for Real-Valued Functions of n Real Variables Partial derivatives: definition, properties, and the concept of the gradient vector. Directional derivatives; directions of maximum and minimum slope. Examples of differentiable but non-continuous functions (for n > 1), and of functions possessing all directional derivatives but not continuous. Differentiability: definition, geometric interpretation, equation of the tangent plane, continuity of differentiable functions, and the total differential theorem. Chain rule for composite functions and gradient formula for expressing the directional derivative of a differentiable function. Higher-order derivatives: definitions, the Hessian matrix, and Schwarz’s theorem. Optimization of Real-Valued Functions of n Real Variables Relative and absolute extrema: first-order necessary condition (Fermat’s theorem) for relative extrema; algorithm for determining the absolute maximum and minimum of a continuous function of two real variables on a compact set. Second-order necessary and sufficient conditions for relative extrema of functions of two real variables: statements, procedure for classifying critical points, analysis of “degenerate” cases where the determinant of the Hessian matrix is zero, and significant examples. Introduction to optimization for functions of three or more real variables. Multiple Integrals Normal domains in the plane: definitions, measure, and properties. Double integrals: introduction, definition, geometric interpretation, and main properties. Reduction formulas for double integrals over normal domains. Change of variables in double integrals, with particular attention to polar coordinates. Notable examples and integration of functions symmetric or antisymmetric with respect to one variable over planar domains symmetric with respect to the coordinate axes. Centroid of planar domains. Elementary notions of triple integrals. Ordinary Differential Equations (ODEs) Notable examples and applied motivations arising from mathematical models in physics and engineering. n-th order ODEs: definition of solution, general integral, normal form, and Cauchy problem. First-order ODEs: theorem of local existence and uniqueness of solutions to the Cauchy problem, with examples and counterexamples; separable equations and the algorithm for finding their solutions. First-order linear ODEs: general integral of homogeneous and non-homogeneous equations. n-th order linear ODEs: local existence and uniqueness theorem for the Cauchy problem, structure of the general integral for the homogeneous and non-homogeneous cases, and the superposition principle. Second-order linear homogeneous ODEs with constant coefficients: characteristic equation and the general solution in the three cases determined by the discriminant (positive, zero, negative). Second-order linear non-homogeneous ODEs with constant coefficients: structure of the general solution and determination of a particular solution using the method of similarity. Linear Differential Forms, Vector Fields, and Line Integrals Line integral of a function (first kind): definition and properties. Definition of an exact Linear Differential Form (LDF) and characterization of primitives of an exact LDF on a connected domain. Conservative vector fields: terminology, physical motivation for LDFs, work of a vector field, and notable examples. Line integral of a Linear Differential Form (second kind). Characterization of exact LDFs. Closed C¹ LDFs: curl and irrotational vector fields. Techniques for computing primitives. Closed LDFs on simply connected open subsets of the plane (with brief remarks for higher dimensions). Local exactness of closed forms. References Primary Textbooks: N. Fusco, P. Marcellini, C. Sbordone, Lezioni di Analisi Matematica Due, Zanichelli, 2020. P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica Due, Prima Parte, Zanichelli, 2018. P. Marcellini, C. Sbordone, Esercitazioni di Analisi Matematica Due, Seconda Parte, Zanichelli, 2018. Additional References: M. Bramanti, C. D. Pagani, S. Salsa, Analisi Matematica 2, Zanichelli, 2009. N. Fusco, P. Marcellini, C. Sbordone, Elementi di Analisi Matematica 2 (Simplified edition for new degree programs), Liguori Editore, 2003. G. Crasta, A. Malusa, Matematica 2, Pitagora Editrice Bologna, 2004.

Prerequisites. The cultural and curricular background required for this course corresponds to the material covered in Mathematical Analysis I. Prior knowledge of Linear Algebra and Geometry will also be beneficial. More specifically, students are expected to have a solid understanding of real-valued functions of a single real variable, including proficiency in computing limits, performing differential and integral calculus, and addressing optimization problems involving such functions.

Analysis II, Terence Tao, Springer

Optional

The written examination, which lasts three hours, is usually divided into four sections, each containing one question assessing theoretical knowledge and one exercise to evaluate acquired skills. The main sections are as follows: ---Properties of n-dimensional Euclidean spaces; parametric curves; introductory concepts on real-valued functions of n real variables: limits and differential calculus; optimization of real-valued functions of two real variables. --Integration: double integrals, line integrals, differential forms, and vector fields. --Power series and Fourier series. --Elements of complex analysis. Students are also provided with mock exams simulating the written test. A student may be exempted from the oral examination if they achieve a passing grade both in the theoretical knowledge section (questions or theoretical exercises) and in the exercises section, and if there is no need for the oral test to allow further in-depth assessment by the instructor. Students may always request to take the oral examination even if they have been exempted. To encourage attendance, the instructor offers the possibility to participate in two partial written exemption tests. The first test is held at the midpoint of the course, covering the topics from the first part, and the second is held at the end of the course. Only students who have passed the first test, even conditionally, are admitted to the second test.

Teaching Methods. A total of 74 hours of lectures and exercises will be delivered using a Flip digital interactive whiteboard, according to the schedule published on the official website of the degree program. Each lecture (in which theoretical knowledge will be presented to students in an interactive and engaging manner) will also include exercise sessions during which students will be guided and “trained” by the instructor in developing the expected skills. The instructor will appropriately balance traditional lecturing and practical exercises in order to maintain a high level of student engagement and attention.