Electronics Engineering

Mathematical Analysis 2 – Course Syllabus Academic Year 2025/26 Prof. Giuseppe Floridia (Provisional version updated on 16/10/2025) n-Dimensional Euclidean Spaces. Properties of 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 sets, compact sets, and connected open sets. Review of Euclidean and analytic geometry in the plane and in space: parallelism and orthogonality of 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 at a point of a curve. Orientation of a curve, admissible reparametrizations, and equivalent curves. Length of a curve and rectifiability of C¹ curves. Notable examples of curves. Introduction to Real-Valued Functions of n Real Variables. Domain and graph of a function. Limits of functions: definitions, properties, and methods for evaluating limits in indeterminate forms using necessary conditions (via path tests) and sufficient conditions (via bounding arguments). Continuous functions: definitions and properties. Absolute maxima and minima, 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 functions differentiable but not continuous (for n > 1) and functions possessing directional derivatives in all directions but not continuous. Differentiability: definition, geometric meaning, tangent plane equation, continuity of differentiable functions, and total differential theorem. Chain rule for composite functions and gradient formula for expressing the directional derivative of a differentiable function. Higher-order derivatives: definitions, Hessian matrix, Schwarz 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 Hessian determinant is zero, and notable 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 notable properties. Reduction formulas for double integrals over normal domains. Change of variables in double integrals, including polar coordinates. Notable examples and integration of functions symmetric or antisymmetric with respect to one variable over planar domains symmetric about the Cartesian axes. Centroid of planar domains. Triple integrals. Integration formulas for wires or layers. Change of variables in triple integrals: cylindrical and spherical coordinates. Centroids, moments of inertia, mass of a body. 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 fields. Techniques for computing primitives. Closed LDFs in simply connected open subsets of the plane (brief remarks on higher dimensions). Local exactness of a closed form. Advanced Integration and Regular Surfaces. Gauss-Green formulas, divergence theorem, and Stokes’ theorem in the plane. Surface integrals of functions. Centroids, moments of inertia, and mass of a surface. Guldino’s theorem for areas of surfaces of revolution. Flux of vector fields through surfaces. Divergence theorem in ℝ³ (without proof). Regular surfaces with boundary. Stokes’ theorem in ℝ³ (without proof). Sequences and Series of Functions. Pointwise and uniform convergence for sequences of functions. Pointwise, uniform, absolute, and total convergence for series of functions. Continuity of the limit function of a sequence or the sum of a series. Passage to the limit under the integral and derivative signs for sequences and series of functions. Power series and Taylor series. Analytic functions. Trigonometric series and Fourier series. Complex Analysis. Introduction to complex numbers; study of power series in the complex field and their properties. Holomorphic functions via the Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, analyticity of holomorphic functions with Goursat’s and Morera’s theorems, Liouville’s theorem, and the fundamental theorem of algebra. Laurent series and classification of isolated singular points, including the Casorati-Weierstrass theorem. Residue theorem for computing complex integrals and study of zeros of holomorphic functions.

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

Examination Methods The written examination, which has a duration of three hours, is usually divided into four sections. Each section includes one question aimed at assessing theoretical knowledge and one exercise designed to evaluate the student’s practical 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 assessment of theoretical knowledge (theoretical questions) and in the practical exercises, and if the instructor deems that an oral examination is not necessary for further clarification or evaluation. However, the student may always request to take the oral examination, even if exempted. To encourage course attendance, the instructor offers students the opportunity to take two partial written tests (midterm exams) as exemptions from the final written examination. The first test takes place midway through the course and covers the topics of the first part of the syllabus, while the second test is held at the end of the course. Only students who have passed the first test—provisionally or definitively—are admitted to the second.

Teaching Methods. A total of 90 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.