Lecturer: Ugo Pietro Gianazza   Marco Veneroni  

Course name: Mathematical Analysis 2
Course code: 500121
Degree course: Ingegneria Edile-Architettura
Disciplinary field of science: MAT/05
L'insegnamento costituisce attività di base per: Ingegneria Edile-Architettura
University credits: ECTS 6
Course website: http://www.imati.cnr.it/~gianazza/anmat2.html

Specific course objectives

The course is the natural prosecution of the Calculus I course, and aims at giving the students, who will not have the possibility to take another Analysis course in the future, a comprehensive expertise of analytical tools, to be used in the more technical courses to come. It does not reduce to a simple recipe book: the focus is on teaching ideas and methods, along with the most significative theorems, all supplied by a large number of examples and exercises, both at introductory and advanced level.

Course programme

1. Power series

• Definition, radius of convergence, properties on the real line.
• Integration and derivation of a power series.
• Taylor series.

2. Multivariate Calculus

• Basic notion of topology and metrics in n-dimensional spaces.
• Continuous functions: properties.
• Partial and directional derivatives; gradient.
• Higher order derivatives.
• Local extrema and main results.
• Vector-valued functions: main properties.

3. Curves

• Definition of regular curve: main properties.
• Rectifiable curves and how to compute their length.
• Arc-length function.
• Arc integrals for real valued functions.

4. Irrotational vector fields

• Arc integral of a vector-valued function.
• Irrotational vector fields: main properties.
• Arc integral of an irrotatioal vector field: the fundamental theorem.
• Conditions for a vector field to be irrotational.

5. Implicit functions

• Implicit function theorem, and regularity of the implicitlt defined function.
• Constrained extrema and the Lagrange multiplier method.

6. Ordinary differential equations

• Existence and uniqueness theorems.
• Linear equations and systems, how to compute the general solution, and how to solve a Cauchy problem.
• A first approach to boundary value problems for simple equations and systems.

7. Multiple integrals

• Definition of a double integral in a rectangle, and how to compute it.
• Extension to a Peano-Jordan measurable set.
• Change of variables.
• Geometric applications.
• Green and divergence theorems for two-variable functions.
• Triple integrals: extension of the methods considered for double integrals.

8. Surfaces

• Regular surfaces: main properties.
• Area of a regular surface.
• Surface integrals and how to compute them.
• Divergence and Stokes theorems for three-variable functions.

Course entry requirements

The student has to master the contents of the Calculus I, Geometry and Linear Algebra courses.

Course structure and teaching

Lectures (hours/year in lecture theatre): 60
Practical class (hours/year in lecture theatre): 20
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

N. Fusco, P. Marcellini, C. Sbordone. Analisi Matematica due. Liguori.

M. Bramanti, C.D. Pagani, S. Salsa. Analisi matematica 2. Zanichelli.

Testing and exams

The final test consists of a written and an oral exam, which have to be taken in the same session.