Mathematical Analysis II

By the end of the course, students are expected to have acquired the following knowledge, skills and competences.

1. Knowledge and understanding

Students will acquire a solid knowledge of the foundations of differential and integral calculus for functions of several real variables, of the theory of ordinary differential equations, of metric spaces, and of Lebesgue measure and integration theory.

2. Applying knowledge and understanding

Students will be able to apply the theoretical techniques and results studied to the solution of problems and exercises, including non-standard ones, concerning the topics of the course.

3. Making judgements

Students will develop the ability to independently assess the correctness and logical coherence of mathematical arguments, identifying the hypotheses required for the application of the theorems studied.

4. Communication skills

Students will be able to present, both in written and oral form, the concepts, proofs and results studied in a clear, rigorous and formally correct way.

5. Learning skills

Students will develop the autonomy required to pursue the study of more advanced topics in Mathematical Analysis and in disciplines that make use of it.

The course is structured into 15 ECTS credits, of which 12 credits (84 hours) are dedicated to lectures and 3 credits (36 hours) to practice sessions.

The concepts and methods covered by the course will be presented through lectures (Direct Teaching). For each topic, the instructor will carry out an adequate number of tutorials. To develop judgement autonomy and communication skills, and to make participation in lectures more active and productive, some hours will be devoted to guided tutorials (Interactive Teaching), in which various exercises will be proposed. Students may work individually or in groups and discuss with one another.

The course is delivered through traditional lectures at the blackboard. If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above, in order to ensure consistency with the syllabus.

A solid command of limits, derivatives and integrals of real functions of a real variable is essential, together with the topics of Linear Algebra and Geometry.

Attendance at lectures is strongly recommended, as well as attendance at  possible tutoring activities.

N.B.: topics marked with an asterisk (*) are to be considered essential minimum knowledge.

1. Sequences and Series of Functions

*Sequences of functions. *Series of functions. *Pointwise, uniform and total convergence. Continuity, term-by-term integration and term-by-term differentiation theorems for series (statements only). *Power series in the real field. *Radius of convergence. D'Alembert's and Cauchy-Hadamard's theorems. *Radius of convergence of the derived series. Differentiation and integration theorems for power series (statements only). *Taylor series. *Criterion for Taylor-series expandability. *Standard series expansions.

2. Real functions of two or more real variables

Elementary topology in Rⁿ. Bounded sets. Connected open sets. *Limits and continuity. Weierstrass' Theorem. *Partial derivatives. Higher-order derivatives. *Schwarz' Theorem (statement only). *Gradient. *Differentiability. *Differentiability and continuity. The differential theorem. *Composite functions. Chain rule for composite functions. *Functions with zero gradient on a connected set. *Local extrema. Necessary and sufficient conditions for a local extremum. Implicit functions. Dini's Theorem. Plane and skew curves. Constrained local maxima and minima. Constraint-explicitation method and the method of Lagrange multipliers. A sufficient condition for a constrained local extremum.

3. Line integrals and differential forms in Rⁿ

*Regular curves. Tangent and normal vector of a regular curve at a point. *Rectifiability. *Length of a regular curve. Oriented curves. Arc-length parameter. Line integral of a function. *Differential forms. *Line integral of a differential form. *Exact differential forms. *Integration theorem for exact differential forms. *Characterization of exact differential forms. *Potential of a differential form. *Closed differential forms. *Differential forms on a simply connected open subset of Rⁿ.

4. Lebesgue measure. Lebesgue integral

Lebesgue measure of rectangles and multi-rectangles in Rⁿ. Measure of non-empty bounded open sets and of sequentially compact sets. Notion of measurability for bounded sets and for not necessarily bounded sets. Properties: monotonicity, finite additivity, subtractivity, σ-additivity, upper and lower continuity, translation invariance, completeness. Measurable functions and their properties. Sequences of measurable functions. Simple functions. Outline of Lebesgue integration theory in Rⁿ. Definition of the integral for measurable, non-negative functions on measurable sets. First properties: monotonicity, isotony, homogeneity. B. Levi's Theorem. Fatou's Lemma. Further properties of the integral: additivity with respect to the integrand function, σ-additivity with respect to the domain of integration, upper and lower continuity, sets of measure zero. Summable functions. Properties. Lebesgue's Theorem. Comparison between the Lebesgue integral and the Riemann integral.

5. Integral Calculus for functions of several variables

Measure and integration on product spaces. Reduction formulas for multiple integrals. Fubini's and Tonelli's Theorems. Change-of-variable Theorems for multiple integrals. Polar coordinates in the plane and in space, cylindrical coordinates.

6. Surfaces. Surface integrals. Stokes' formula and the Divergence Theorem

Regular surfaces in Rⁿ. Tangent plane and normal vector. *Gauss-Green Theorem. *Divergence Theorem. Stokes' formula. Area of a surface, surface integrals, flux of a vector field through a surface, quadratic differential forms.

7. Theory of ordinary differential equations

Definitions and terminology. Peano's existence Theorem. *Local existence and uniqueness Theorem for the Cauchy problem associated with a first-order differential equation in normal form. Global existence and uniqueness Theorem, maximal solutions. *First-order separable, homogeneous, linear, Bernoulli, exact and Clairaut differential equations. *Linear differential equations: structure of the solution set, method of variation of arbitrary constants. *Linear differential equations with constant coefficients. Euler's equations. *Linear systems of differential equations. *Exponential matrix.

8. Metric spaces

Metric spaces. Neighborhoods. Interior, boundary and accumulation points. Derived set. Open, closed and bounded sets. Diameter. Convergence. Completeness. Lipschitz functions and the Contraction Theorem. Sequential compactness. Closure and its characterization via sequences. Cantor's Theorem. Euclidean space Rⁿ. Cauchy-Schwarz and Minkowski inequalities. Heine-Borel and Bolzano-Weierstrass Theorems. Convex, connected and polygonally connected sets.

9. Elements of Calculus of Variations and Partial Differential Equations

The brachistochrone problem. Functionals. Euler-Lagrange equation. Necessary conditions for an extremum. Examples and applications. Outline of first- and second-order partial differential equations.

6.                         M.Giaquinta - G. Modica Mathematical Analysis: An Introduction to Functions of several variables Birkhäuser 

7.                         W. Rudin - Principles of Mathematical Analysis 3 ed Mc Graw Hill

1. The final examination consists of a written test divided into two sections — Section A (practical and theoretical questions on the syllabus covered in the first teaching period) and Section B (practical questions on the syllabus covered in the second teaching period) — and an oral examination on the part of the syllabus taught during the second teaching period.
2. At the end of the first teaching period, a midterm written exam is offered with theoretical and practical questions on the topics covered in that period.
3. Passing the midterm exam, with a minimum score of 18/30, exempts the student from completing the questions in Section A of the final examination.
4. The benefits of passing the midterm exam remain valid until the end of the third examination session of the current Academic Year.
5. Students who have not taken or have not passed the midterm exam must complete the entire written test (Sections A and B) in the final examination. The written exam is considered passed if a minimum score of 18/30 is achieved in each of the two sections.

PLEASE NOTE: the final grade will take into account correctness, rigor and clarity of exposition, in both the written and the oral examination.

●      30 cum laude / 30: excellent performance, full command of the topics, ability to connect concepts and complete rigor in exposition.

●      26-29: good/very good performance, with minor inaccuracies.

●      21-25: fair performance, with some non-serious gaps.

●      18-20: sufficient performance, with knowledge of the essential topics.

●      Failed: insufficient performance, serious and/or widespread gaps.

Learning assessment may also be carried out on-line, should the conditions require it.

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l'integrazione Attiva e Partecipata — Servizi per le Disabilità e/o i DSA) referring teacher within their department (https://www.cinap.unict.it/content/referenti).

●      Differential forms.

●      Relationship between the existence of partial derivatives (derivability) and differentiability for a function of two or more real variables.

●      Constrained extrema of a function.

●      Gauss-Green Theorem.

●      Lebesgue measure.