Mathematics II

Knowledge and understanding: students will learn some basic mathematical concepts and will develop the skills of calculation and manipulation of the most common objects of Mathematical Analysis: among these, integrals for functions of one or several real variables, the differential equations and the differential calculus for real functions of two or several real variables.

Applying knowledge and understanding: using examples related to applied sciences, the student will be able to appreciate the importance of Mathematical Analysis as an important modeling tool.

Making judgments: students will be able to deal with some simple but significant methods of Mathematical Analysis with sufficient rigor to refine their logical skills. Many demonstrations will be presented in a schematic and intuitive way to engage students and encourage them to achieve the goal on their own.

Communication skills: by studying Mathematical Analysis, students will learn to communicate with rigor and clarity both orally and in written form. They will learn that using a correct language is one of the most important means of clearly communicating any scientific topic, not only mathematics.

Learning skills: students, especially the most willing, will be stimulated to deepen some topics, also through group work.

Information for students with disabilities and/or DSA: to guarantee equal opportunities and in compliance with the laws in force, interested students can request a personal  interview in order to plan any compensatory and/or dispensatory measures, based on the educational objectives and specific needs.

In this case, it is advisable to contact the CInAP (Centre for Active and Participated Integration - Services for Disabilities and/or SLD) professor of the Department where the Degree Course is included.

Lectures complemented by exercises

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.

Exams may take place online, depending on circumstances.

Integral calculus for functions of one variable

Indefinite integrals and their properties – Methods of integration: integration by decomposition and linearity, integration of rational functions, integration by parts, integration by substitution – Definition of the Riemann integral and its properties – Some classes of integrable functions – Definite integrals – Geometric interpretation of the Riemann integral – Fundamental Theorem of Calculus – Introduction to generalized and improper integrals and their properties.

Differential calculus for functions of two or several variables

Taylor polynomial of a real-valued function of one real variable – Review of topology in the plane: interior, exterior and boundary points; open and closed sets; accumulation points and isolated points; bounded sets; compact sets; convex sets; path-connected sets; domains – Functions of several variables: limits and continuity; Weierstrass theorem – Differential calculus for functions of several variables: partial and directional derivatives – Differential and differentiable functions – Higher-order derivatives and Schwarz's theorem – Differential operators: gradient, divergence, curl, Laplacian – Chain Rule – Lagrange's Theorem in R^2 and characterization of functions with zero gradient on a region – Unconstrained extrema of functions of two variables and related theorems – Finding absolute extrema on a compact set.

Ordinary differential equations

Generalities on differential equations – The Cauchy problem – First-order differential equations – First-order separable differential equations – Cauchy's existence and uniqueness theorem – Second-order linear differential equations with constant coefficients – Applications to mathematical models.

Introduction to the geometry of curves and on linear differential forms

Regular and piecewise regular curves – Rectifiable curves and their length – Arc length – Line integrals of functions – Linear differential forms – Line integrals of linear differential forms – Exact and closed differential forms – Applications.

1. C. Canuto, A. Tabacco – Mathematical Analysis I – Springer-Verlag Italia
2. C. Canuto, A. Tabacco – Mathematical Analysis II – Springer-Verlag Italia

The final examination consists of a written test divided into two parts. The first part covers definitions and theorems to be stated and/or proved, together with relevant examples. The second part consists of open-ended exercises covering the main topics included in the course syllabus. The final examination may also include an oral examination. Both the written test and the oral examination are graded on a scale of 30.

N.B.: The assessment of learning outcomes may also be conducted remotely, if required by the circumstances.

The written and/or oral examination will cover the following topics:

Definitions:

Define the limit of functions of several variables; state the theorems concerning limits of functions; define continuous functions and discuss their properties; define partial and directional derivatives, differentiable functions and their properties, and differential forms.

Theorems – statement and proof (when covered in class):
State Schwarz's theorem. State and prove the Fundamental Theorem of Calculus.

Examples and/or counterexamples illustrating properties of functions or theorems:
Provide examples and/or counterexamples illustrating properties of functions or theorems, such as the possible non-differentiability of continuous functions, and the relationship between a relative extremum of a function and the vanishing of its gradient at that point.