METODI ANALITICI PER L'INGEGNERIA II

Knowledge and understanding

By the end of the course, students will be familiar with the fundamental concepts and methods of differential and integral calculus for functions of several real variables, the basic tools for the study of sequences and series of functions, the differential geometry of curves and surfaces, the calculus of linear differential forms together with the Gauss-Green and Stokes theorems, as well as the fundamental techniques for the study of ordinary differential equations. Students will also become familiar with the rigorous language of mathematical analysis and with the main proof techniques used in the discipline.

Applying knowledge and understanding

By the end of the course, students will be able to: compute partial derivatives and study the differentiability of functions of several variables; determine free and constrained maxima and minima; compute multiple integrals, including by means of changes of variables; study the pointwise and uniform convergence of sequences and series of functions; parametrise curves and surfaces and compute their length and area; check whether a differential form is exact and, if so, compute a potential for it; apply the Gauss-Green and Stokes formulas to the computation of line and surface integrals; set up and solve Cauchy problems for ordinary differential equations, in particular linear equations with constant coefficients. These skills provide the foundation for modelling and solving problems typical of building engineering and architecture.

The course consists of lectures, during which the theoretical content of the course is presented and discussed rigorously, and tutorials, devoted to the guided solution of applied exercises and to problems representative of the types that will be the subject of the examination.

Taught activities (DE – Didattica Erogativa) include the presentation of theoretical content and applied examples; interactive activities (DI – Didattica Interattiva) include classroom tutorials, with the active involvement of students in solving exercises and discussing questions proposed by the lecturer, also for the purpose of ongoing self-assessment.

Teaching activities are organised so as to ensure consistency between the learning objectives and the methods adopted: the frontal presentation of theoretical content is constantly accompanied by applied tutorials, aimed at developing the computational and problem-solving skills required by the intended learning outcomes.

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.

In order to follow the course profitably, students must have a solid command of differential and integral calculus for functions of one real variable (limits, continuity, differentiation, function analysis, Riemann integration, numerical series) and of the basic elements of geometry and linear algebra (vectors, matrices, linear systems, analytic geometry of the plane and of space).

It is also important to have a good command of algebraic and trigonometric calculus and of the basic concepts of the topology of the real line. Such knowledge is normally provided in the introductory courses of Mathematical Analysis I and Geometry offered in the first year of engineering degree programmes.

Attendance is mandatory, in accordance with the rules set out in the CdS (Degree Programme) teaching regulations.

1.    Differential calculus in several variables. 

Partial derivatives; differentiable functions; higher-order derivatives; composite functions; relative maxima and minima.

2.    Integral calculus in several variables. 

Integration; measure of sets; integrability of continuous functions; computation of double integrals; volume of solids; change of variables; polar coordinates; improper integrals.

3.    Sequences and series of functions. 

Sequences of functions; uniform convergence; series of functions; power series.

4.    Curves and surfaces. 

Curves in ℝⁿ; length of a curve; surfaces; area of a surface; the implicit function theorem; constrained maxima and minima.

5.    Differential forms. 

Differential forms; exact forms; the Gauss-Green formula; applications of the Gauss-Green formula; the Stokes formula.

6.    Differential equations. 

•      Enrico Giusti, “Analisi Matematica II”, Bollati Boringhieri.

•      P. Marcellini, C. Sbordone, “Esercizi di Analisi Matematica 2”, Zanichelli.

•      S. Salsa, A. Squellati, “Esercizi di Analisi Matematica”, Zanichelli.

Any lecture notes and supplementary teaching material prepared by the lecturer will be made available during the course.

The exam consists of:

• Written test – prerequisite for the oral exam

• Oral exam

Grading Criteria

Grade 29–30 with honors

• In-depth knowledge of the subject

• Ability to integrate and critically analyze the presented situations

• Independent resolution of complex problems

• Excellent communication skills and command of language

Grade 26–28

• Good knowledge of the subject

• Ability to analyze situations critically and coherently

• Fairly independent resolution of complex problems

• Clear presentation with appropriate language

Grade 22–25

• Satisfactory knowledge, limited to main topics

• Critical analysis not always coherent

• Presentation fairly clear with adequate command of language

Grade 18–21

• Minimal knowledge of the subject

• Limited ability to integrate and critically analyze situations

• Presentation sufficiently clear, but language skills poorly developed

Exam not passed

• Insufficient knowledge of the main course contents

• Very limited or no ability to use specific terminology

• Unable to independently apply acquired knowledge

Compensatory and Dispensatory Measures

To ensure equal opportunities and comply with current regulations:

• Interested students may request a personal meeting to arrange any compensatory and/or dispensatory measures, according to the learning objectives and specific needs.

• Students may also contact the CInAP representative (Centre for Active and Participatory Integration – Services for Disabilities and/or Learning Disorders) of their Department.

•      Determine the free extrema of the function f(x, y) and classify them by means of the Hessian matrix.

•      Determine the constrained extrema of a function on a set defined by an equation, using the method of Lagrange multipliers.

•      Compute a double integral over a given domain, possibly by means of a change of variables (polar coordinates).

•      Study the pointwise and uniform convergence of a given sequence or series of functions.

•      Determine the radius and the set of convergence of a power series.

•      Check whether a linear differential form is exact and, if so, compute a potential for it.

•      Compute the length of a curve or the area of a surface by means of a parametrisation.

•      Apply the Gauss-Green formula to compute a line integral or a plane area.

•      Solve a Cauchy problem for a linear differential equation, including with constant coefficients.

•      State and prove one of the fundamental theorems of the course (e.g. the implicit function theorem, the Stokes formula, the existence and uniqueness theorem for the Cauchy problem).