Mechanical Engineering

Program of Mathematical Analysis, course II (9 credits), Prof. L. Moschini Bachelor degree in Mechanical Engineering SEQUENCES AND SERIES OF FUNCTIONS: Point wise and uniform convergence of sequences of functions, Point wise, uniform, absolut and total convergence of series of functions. Continuity of the limit function or of the sum of a serie. Integration and differentiation of functional sequences and series. Abel’s theorem (without proof). Power and trigonometric series. Taylor and Fourier’s series. FUNCTIONS OF MULTIPLE VARIABLES. General background on the vector space R^n. Topological properties of R^n. Limits and continuity. Theorems on continuous functions. Weierstrass’s theorem (without proof). Uniform continuity. Heine-Cantor’s theorem (without proof). Partial derivatives, directional and successive derivatives. Schwartz’s theorem (without proof). Differentiability. Total differential theorem. The gradient formula. The derivation under the integral sign. Taylor’s development up to the second order with Lagrange and Peano remainder. Critical points. Relative maxima and minima for C^2 functions. Searching for absolute maximum and minimum. Homogeneous functions. LINEAR DIFFERENTIAL FORMS: Curves in R^n. Length of a regular curve and curvilinear abscissa. The curvilinear integral of a function. Exact and closed differential forms. The curvilinear integral of a linear differential form. MULTIPLE INTEGRALS: Normal domains in the plane. Integrability of continuous functions. Reduction formula for double integrals (without proof). Guldino’s theorem for the volume of a solid of revolution. Gauss-Green’s formula. The divergence theorem on the plane. Stokes’s formula on the plane. Changes of variables in double integrals: the case of polar coordinates. Triple integrals and their reductions formulas. Changes of variables in triple integrals: the case of spherical and cylindrical coordinates. Mass, inertia and center of mass of a solid. REGULAR SURFACES: Surface integrals. Mass, inertia and center of mass of a surface. Flux of a vector field through a surface. Guldino’s theorem for the area of a surface of revolution. Divergence’s theorem (without proof). Regular surfaces with edge and Stokes’s formula (without proof). IMPLICIT FUNCTIONS: Dini’s theorem. Conditional maxima and minima: Lagrange multipliers.

Mathematical analysis 1. Fundamental elements of the real functions of one real variable (limits and continuity, differential calculus and optimization, integral calculus), sequences and series of real numbers.

Textbook “Lezioni di Analisi Matematica II” written by L. Moschini, edited by Esculapio, Bologna. ISBN 978-88-9385-278-4 Excercise book “Esercizi svolti di Analisi Matematica II” written by L. Moschini, edited by Esculapio, Bologna. ISBN 978-88-9385-279-1

Optional

The exam consists of a written test and, at discretion of the examiner or the student, an oral examination. The written part consists of three exercises and nine true/false questions whose answer needs to be explained. The exercises and true/false questions are basse on the course material. Each exercise is worth a maximum of 7 points. Each correct true/false answer without an explanation or with an incorrect explanation is worth 0,5 points, while each correct true/false answer with a correct explanation is worth 1 point. Each incorrect true/false answer is worth -0,5 points, and each unanswered true/false question is worth 0 points. A student with a score between 16 and 17,5 is always allowed to supplement their written exam during the oral exam to reach a passing grade.

SEQUENCES AND SERIES OF FUNCTIONS: Point wise and uniform convergence of sequences of functions, Point wise, uniform, absolut and total convergence of series of functions. Continuity of the limit function or of the sum of a serie. Integration and differentiation of functional sequences and series. Abel’s theorem (without proof). Power and trigonometric series. Taylor and Fourier’s series. FUNCTIONS OF MULTIPLE VARIABLES. General background on the vector space R^n. Topological properties of R^n. Limits and continuity. Theorems on continuous functions. Weierstrass’s theorem (without proof). Uniform continuity. Heine-Cantor’s theorem (without proof). Partial derivatives, directional and successive derivatives. Schwartz’s theorem (without proof). Differentiability. Total differential theorem. The gradient formula. The derivation under the integral sign. Taylor’s development up to the second order with Lagrange and Peano remainder. Critical points. Relative maxima and minima for C^2 functions. Searching for absolute maximum and minimum. Homogeneous functions. LINEAR DIFFERENTIAL FORMS: Curves in R^n. Length of a regular curve and curvilinear abscissa. The curvilinear integral of a function. Exact and closed differential forms. The curvilinear integral of a linear differential form. MULTIPLE INTEGRALS: Normal domains in the plane. Integrability of continuous functions. Reduction formula for double integrals (without proof). Guldino’s theorem for the volume of a solid of revolution. Gauss-Green’s formula. The divergence theorem on the plane. Stokes’s formula on the plane. Changes of variables in double integrals: the case of polar coordinates. Triple integrals and their reductions formulas. Changes of variables in triple integrals: the case of spherical and cylindrical coordinates. Mass, inertia and center of mass of a solid. REGULAR SURFACES: Surface integrals. Mass, inertia and center of mass of a surface. Flux of a vector field through a surface. Guldino’s theorem for the area of a surface of revolution. Divergence’s theorem (without proof). Regular surfaces with edge and Stokes’s formula (without proof). IMPLICIT FUNCTIONS: Dini’s theorem. Conditional maxima and minima: Lagrange multipliers.

Mathematical analysis 1. Fundamental elements of the real functions of one real variable (limits and continuity, differential calculus and optimization, integral calculus), sequences and series of real numbers.

M. Bertsch, A. Dall'Aglio, L. Giacomelli: Epsilon 2, McGraw Hill (2024) Lezioni di Analisi Matematica 2, Luisa Moschini, editrice Esculapio Bologna

Lessons in classroom

The exam consists of a written test and, at the discretion of the examiner or the student, an oral examination. The written part consists of three exercises and nine true/false questions whose answer needs to be explained. The exercises and true/false questions are based on the course material. Each exercise is worth a maximum of 7 points. Each correct true/false answer without an explanation or with an incorrect explanation is worth 0.5 points, while each correct true/false answer with a correct explanation is worth 1 point. Each incorrect true/false answer is worth -0.5 points, and each unanswered true/false question is worth 0 points. A student with a score between 16 and 17.5 is always allowed to supplement their written exam during the oral exam to reach a passing grade.

logna

electronic blackboard and/or traditional blackboard and/or writing tablet