MATHEMATICAL ANALYSIS
Stampa
Enrollment year
2021/2022
Academic year
2021/2022
Regulations
DM270
Department
DEPARTMENT OF CIVIL ENGINEERING AND ARCHITECTURE
Course
CIVIL AND ENVIRONMENTAL ENGINEERING
Curriculum
PERCORSO COMUNE
Year of study
Period
Annual (27/09/2021 - 17/06/2022)
ECTS
12
Language
Italian
Prerequisites
Mathematics: the required prerequisites for enrollment in the School of Engineering.
Learning outcomes
The course is aimed at providing the basic knowledge of calculus (differential,
integral) for real and vector-valued functions of one or several real variables,
together with the fundamentals of the theory of series and with an introduction to ordinary differential equations. Lectures will be mainly focused on the comprehension of notions (definitions, results), although some proofs will still be detailed. Examples and exercises will be presented. By the end of the course the
Students are expected to be able to handle correctly and without hesitation limits,
derivatives, function graphs, integrals (also multiple, line and surface integrals), series, linear differential equations, and the corresponding theoretical facts.
Course contents
Functions, Limits, Continuity.
Recalls and complements on real numbers. Complex numbers. Functions:
definitions, graphs; invertible functions; odd and even functions; monotone
functions; periodic functions; operations on functions; nested functions. Elementary
functions and corresponding graphs. Limits of functions: definitions, operations on
limits. Continuous functions. Discontinuity points and their classification. Global
properties of continuous functions.

Differential Calculus in one real variable and Applications.
Derivative of a function: definition and properties, applications in Geometry and
Physics. Derivation rules and calculus. Fundamental theorems of differential
calculus. Primitives and indefinite integrals. Successive derivatives. Function study:
extrema, monotonicity, convexity. De l'Hopital rules.

Integral Calculus in one real variable and Applications.
Definite integrals: definitions and basic properties, applications in Geometry and
Physics. Fundamental theorems of integral calculus. Integration techniques.
Improper integrals.

Series. Limits of real sequences. Real series: definitions and basic examples; series with
positive terms (and convergence tests); absolute and simple convergence.
Fundamentals of real power series. Taylor polynomials and Taylor formulas. Taylor
series; Taylor series of some elementary functions.

Differential Equations.
Introduction to ordinary differential equations. The Cauchy problem. Linear ordinary differential equation of the first order, separable equations, homogeneous
equations. Linear ordinary differential equation of higher order with constant
coefficients: homogeneous and complete cases. Fundamentals about boundary value problems for second order equations.

Differential Calculus in several real variables. Real functions of several real variables: definitions, graphs; limits and continuity; partial derivatives, gradients, and directional derivatives. Successive derivatives.
Differentiability. Partial derivatives of nested functions (chain rules). Free extrema of
real functions of several real variables; critical points and their classification.
Fundamentals of differential calculus for vector-valued functions; jacobian matrices.

Multiple Integrals.
Double integrals: definitions and basic properties, application in Geometry and Physics. Integration techniques: iteration formulas; change of variables; double integrals in polar coordinates. Fundamentals of volume integrals.

Line Integrals and Surface Integrals. Curves: tangent vectors; rectifiable curves and arc length. Surfaces: tangent planes; surface area. Line integrals with respect to the arc length. Line integrals of vector fields, and applications in Physics. Gradient fields, potentials, and path independence. The operators curl and div. Surface integrals, and applications
in Physics. Green's theorem and divergence theorem in two variables. Stokes'
theorem and divergence theorem in three variables.
Teaching methods
Lectures (hours/year in lecture theatre): 46
Practical class (hours/year in lecture theatre): 74
Practicals / Workshops (hours/year in lecture theatre): 0
Reccomended or required readings
M. Bramanti, C.D. Pagani e S. Salsa. Matematica. Calcolo infinitesimale e Algebra
lineare (seconda edizione). C.E. Zanichelli, Bologna, 2004. (Recommended book).

M. Bramanti, C.D. Pagani e S. Salsa . Analisi Matematica 1 (prima edizione) e
Analisi Matematica 2 (prima edizione). C.E. Zanichelli, Bologna, 2008-2009.
Assessment methods
The exam consists of a written examination and an optional oral examination: the written examination is further divided into two parts: exercises (first part) and theoretical questions (second part). Written and optional oral examinations should be passed within the same session. The oral examination is based on definitions, examples and counterexamples, theorems (some with proofs).

For more information see the web-page: http://matematica.unipv.it/rocca/


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