Lecturer: Ugo Pietro Gianazza  

Course name: Analisi matematica 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: CFU 6
Course website: n.d.

Specific course objectives

This course naturally complements the Mathematical Analysis I course the students took the previous year. Since the students will not take further Analysis courses in their academic career, here the aim is to provide them with a reasonably complete background on the main analytical tools. Things are not reduced to a simple laundry list: we try and give the main ideas, the most significant theorems, together with a list of fully developed, introductory examples.

Course programme

Power Series

• Definition, radius of convergence, main properties in R
• Power series derivation and integration
• Taylor series

Functions of Several Real Variables

• Elements of Topology in N-dimensional spaces
• Continuous functions: main properties
• Partial and directional derivatives; the gradient
• Higher order derivatives
• Local minima and maxima: main theorems
• Vectorial functions: main properties

Curves

• Definition of regular curve: main properties
• Rectifiable curves and their length
• Arc length
• Line integral

Conservative Vector Fields

• Line integral of a vector-valued function
• Conservative vector fields: main properties
• Line integral of a conservative vector field: main theorem
• Conditions for a vector field to be conservative

Implicit Functions

• Implicit function theorem: existence and regularity of the implicitly defined function
• Lagrange multipliers for local, constrained minima and maxima

Ordinary Differential Equations

• Existence and uniqueness theorems
• Linear equations: the Cauchy problem
• Boundary value problems and systems

Multiple Integrals

• Double integral in a square
• Extension to Peano-Jordan measurable sets
• Change of variables
• Geometric applications
• Green and divergence theorems in the plane Triple integrals: extension of the previous concepts

Surfaces

• Regular surfaces: main properties
• Area of a surface
• Surface integrals
• Stokes and divergence theorems in R^3

Course entry requirements

All the notions the student has previously seen in the Geometry and Mathematical Analysis I 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.

Appunti del docente.

Testing and exams

The exam consists in a written and in an oral test on all the course topics.