Enrollment year
2019/2020
Academic discipline
MAT/05 (MATHEMATICAL ANALYSIS)
Department
DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING
Course
ELECTRONIC AND COMPUTER ENGINEERING
Curriculum
PERCORSO COMUNE
Period
2nd semester (02/03/2020 - 12/06/2020)
Lesson hours
83 lesson hours
Activity type
WRITTEN AND ORAL TEST
Prerequisites
Calculus I, Geometry and Linear Algebra.
Learning outcomes
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 contents
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.
Teaching methods
Lectures (hours/year in lecture theatre): 60
Practical class (hours/year in lecture theatre): 30
Practicals / Workshops (hours/year in lecture theatre): 0
Reccomended or required readings
M. Bramanti, C.D. Pagani, S. Salsa. Analisi Matematica 2. Zanichelli, Bologna, 2009.
Assessment methods
The final test consists of a written and an oral exam, which have to be taken in the same session.
Further information
The final test consists of a written and an oral exam, which have to be taken in the same session.
Sustainable development goals - Agenda 2030
Università di Pavia - Offerta formativa