Lecturer: Ugo Pietro Gianazza   Marco Veneroni  

Course name: Mathematical analysis 2
Course code: 500121
Degree course: Bioingegneria, Ingegneria Elettronica e Informatica
Disciplinary field of science: MAT/05
L'insegnamento costituisce attività di base per: Bioingegneria, Ingegneria Elettronica e Informatica
University credits: ECTS 9
Course website: n.d.

Specific course objectives

The course is the natural prosecution of the Calculus I course, and aims at giving the students a comprehensive expertise of analytical tools, to be used in the more technical courses to come. The students will learn how to handle scalar- and vector-valued functions depending on several variables, compute partial derivatives, evaluate multiple integrals and integrals along lines and on surfaces. Besides the most significant theorems on the topic, stated with mathematical rigor, a large number of examples and exercises will be provided in order to teach methods and ideas.

Course programme

Power series

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

Multivariate Calculus

• Basic notion of topology and metrics in n-dimensional spaces.
• Continuous functions.
• Partial and directional derivatives; differentiability.
• Higher order derivatives.
• Optimization and main results.
• Vector-valued functions.

Curves

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

Multiple integrals

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

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.

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.

Course entry requirements

Calculus I, Geometry and Linear Algebra.

Course structure and teaching

Lectures (hours/year in lecture theatre): 90
Practical class (hours/year in lecture theatre): 0
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

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.