Enrollment year
2018/2019
Academic discipline
MAT/05 (MATHEMATICAL ANALYSIS)
Department
DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING
Curriculum
PERCORSO COMUNE
Period
2nd semester (06/03/2019 - 14/06/2019)
Lesson hours
83 lesson hours
Activity type
WRITTEN AND ORAL TEST
Prerequisites
This course is part of the basic mathematical training of Engineering students. In order to fruitfully follow this course, the students should have followed the basic courses: 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.
Vector fields:
Arc integral of a vector-valued function. Irrotational vector fields: main properties. Arc integral of an irrotational vector field: the fundamental theorem.
Conditions for a vector field to be irrotational.
Teaching methods
Lectures (hours/year in lecture theatre): 45
Practical class (hours/year in lecture theatre): 38
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.
S. Salsa e A. Squellati. Esercizi di Analisi Matematica 2. Zanichelli, Bologna, 2011.
Assessment methods
The examination is composed by a mandatory written test and an optional oral test. In the written test the students are requested to solve exercises (1st part) and answer theory questions (2nd part). The oral examination, which has to be carried out in the same session of the written one, will test theorems' statements and proofs, definitions, and fundamental examples and counterexamples.
Further information
The examination is composed by a mandatory written test and an optional oral test. In the written test the students are requested to solve exercises (1st part) and answer theory questions (2nd part). The oral examination, which has to be carried out in the same session of the written one, will test theorems' statements and proofs, definitions, and fundamental examples and counterexamples.
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