Ingegneria Civile per l'Ambiente ed il Territorio | MATHEMATICS II

cod. 0612500002

MATHEMATICS II

	0612500002
	DIPARTIMENTO DI INGEGNERIA CIVILE
	EQF6
	CIVIL AND ENVIRONMENTAL ENGINEERING
	2018/2019

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2018
	SECONDO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/05	9	90	LESSONS	BASIC COMPULSORY SUBJECTS

	Objectives
	THE AIM OF THE COURSE IS THE deeper LEARNING OF THE BASIC CONCEPTS OF MATHEMATICAL ANALYSIS, CALCULUS and geometry, and of the basic tools of linear algebra, with their physical and engineering applications. 1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FINALIZED TO PROVIDE THE STUDENTS WITH a more rich mathematical language a deeper knowledge of the basic mathematical concepts and their representation with particular attention to: matrices and linear systems; vector and euclidean space; eigenvalues and diagonalization; analytic geometry in the space; functions sequences and serie; functions of more variables; differential equations; curves and curvilinear integrals; differential forms and integral on paths; multiple integrals; surfaces and surface integrals. 2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WILL BE ABLE TO FORMULATE MATHEMATICALLY AND TO SOLVE SIMPLE PROBLEMS OF THE APPLIED SCIENCES, EXPECIALLY IN THE CIVIL ENGINEERING FRAMEWORK. IN PARTICULAR, THE STUDENT WILL BE ABLE TO use vector and matrix calculus, tosolve linear systems, compute eigenvalues and eigenvectors, describe analytically and represent geometric figures in the space, establish the convergence of functions sequences and series and compute simple sums, use the differential calculus of more variables, solve problems of maximum and minimum and differential equations, compute the length of a curve and curvilinear integrals of functions and differential forms, multiple integrals, areas and surface integrals. 3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO CHOOSE THE MOST APPROPRIATE MATHEMATICAL MODEL AND METHOD IN DIFFERENT SITUATIONS, TO ESTABLISH THE PLAUSIBILITY OF A RESULT AND TO CHECK ITS VALIDITY. 4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO EXPRESS WITH THE SUITABLE TECHNICAL LANGUAGE AND TO REPRESENT GRAPHICALLY THE LEARNED MATHEMATICAL NOTIONS AND TECHNIQUES, AND TO INTEGRATE THEM WITH THOSE ONES OF OTHER SCIENTIFIC DISCIPLINES. 5. LEARNING SKILLS: THE STUDENT WILL GET A MATHEMATICAL BACKGROUND WHICH ALLOWS HIM TO LEARN MORE ADVANCED MATHEMATICAL CONCEPTS AND MORE GENERALLY SCIENTIFIC SUBJECTS WHICH USE MATHEMATICAL TOOLS.

THE AIM OF THE COURSE IS THE deeper LEARNING OF THE BASIC CONCEPTS OF MATHEMATICAL ANALYSIS, CALCULUS and geometry, and of the basic tools of linear algebra, with their physical and engineering applications.
1. KNOWLEDGE AND UNDERSTANDING: THE COURSE IS FINALIZED TO PROVIDE THE STUDENTS WITH a more rich mathematical language a deeper knowledge of the basic mathematical concepts and their representation with particular attention to: matrices and linear systems; vector and euclidean space; eigenvalues and diagonalization; analytic geometry in the space; functions sequences and serie; functions of more variables; differential equations; curves and curvilinear integrals; differential forms and integral on paths; multiple integrals; surfaces and surface integrals.
2. APPLYING KNOWLEDGE AND UNDERSTANDING: THE STUDENT WILL BE ABLE TO FORMULATE MATHEMATICALLY AND TO SOLVE SIMPLE PROBLEMS OF THE APPLIED SCIENCES, EXPECIALLY IN THE CIVIL ENGINEERING FRAMEWORK. IN PARTICULAR, THE STUDENT WILL BE ABLE TO use vector and matrix calculus, tosolve linear systems, compute eigenvalues and eigenvectors, describe analytically and represent geometric figures in the space, establish the convergence of functions sequences and series and compute simple sums, use the differential calculus of more variables, solve problems of maximum and minimum and differential equations, compute the length of a curve and curvilinear integrals of functions and differential forms, multiple integrals, areas and surface integrals.

3. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO CHOOSE THE MOST APPROPRIATE MATHEMATICAL MODEL AND METHOD IN DIFFERENT SITUATIONS, TO ESTABLISH THE PLAUSIBILITY OF A RESULT AND TO CHECK ITS VALIDITY.
4. COMMUNICATION SKILLS: THE STUDENT WILL BE ABLE TO EXPRESS WITH THE SUITABLE TECHNICAL LANGUAGE AND TO REPRESENT GRAPHICALLY THE LEARNED MATHEMATICAL NOTIONS AND TECHNIQUES, AND TO INTEGRATE THEM WITH THOSE ONES OF OTHER SCIENTIFIC DISCIPLINES.
5. LEARNING SKILLS: THE STUDENT WILL GET A MATHEMATICAL BACKGROUND WHICH ALLOWS HIM TO LEARN MORE ADVANCED MATHEMATICAL CONCEPTS AND MORE GENERALLY SCIENTIFIC SUBJECTS WHICH USE MATHEMATICAL TOOLS.

	Prerequisites
	Functions of one real variable and analytic geometry of the plane. Derivative and integral of powers, also with fractional exponent, and of rational, trigonometric, exponential and logarithmic functions. Derivation rules and integration techniques. Numerical series and improper integrals.

	Contents
	1. Vector spaces and Euclidean spaces: Vector spaces and subspaces. Linear independence. Basis and dimension. Application to linear systems – Euclidean vector spaces. Scalar product, norms and angles. Cauchy-Schwarz inequality. Orthogonality (h Lectures/Exercises 6/4) 2. Matrices and linear sistems: Matrices. Determinant and rank – Linear systems. Theorems of Cramer and Rouché-Capelli. Gauss method – Eigenvalues and eigenvectors. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizzation (8/6) 3. Analytic geometry of the space: Review of analytic geometry of the pane – Cartesian coordinates in the space. Equation of planes and lines. Vector product. Parallelism and ortogonality conditions (2/4) 4. Sequences and series of functions: Review of numerical series – Sequences of functions. Pointwise and uniform convergence. Continuity of uniform limit. Interchanging limit with integral and derivative sign – Series of functions. Uniform and total convergence. Integration by series. Derivative term by term – Power series. Convergence interval and radius. Analiticity. Taylor series (5/3) 5. Functions of more variables: Basic topology in IRn. Limits and continuity – Partial derivatives. Higher order derivatives. Schwarz Theorem – Gradient. Differentiability. Composite functions. Directional derivatives. Level curves. Functions with zero gradient in a convex domain – Taylor formula and second order differentiability. Hessian matrix. Quadratic forms. Matrices with definite sign. Local and global maxima and minima – Vector valued functions and Jacobian matrix (5/5) 6. Differential equations: General solution and Cauchy problem – Equivalence between equations of order n and systems of n first order equations – The Cauchy theorem on local existence and uniqueness with consequences. Global existence. Maximal solution – Equations with separable variables – Linear differential equations. Wronskian and linear independence. Structure of the general integral of linear homogeneous and non-homogeneous differential equations with constant coefficients. Bernoulli equations. Higher order linear homogeneous and non-homogeneous differential equations with constant coefficients – Linear systems (6/6) 7. Curves and curvilinear integrals: Regular curves. Direction and tangent vector. Curve orientation – Length of a curve. Rectifiability. Piecewise regular curves. Curvilinear integral of a function (3/2) 8. Differential forms: Vector fields. Work. Conservative fields – Linear differential forms. Curvilinear integral of a differential form. Exact differential forms. Primitives and potential of the associated vector field – Exactness criterion in connected domains. Closed forms. Exactness criterion in star-shaped domains (4/3) 9. Multiple integrals: Double integrals over normal domains. Reduction formulas – Gauss-Green formulas in the plane. Divergence theorem. Stokes formula – Changing variables. Polar coordinates – Triple and higher dimensional integrals (5/7) 10. SURFACE AND SURFACE INTEGRALS: Regular surfaces. Tangent plane and normal vector – Surface area – Oriented surfaces. Boundary – Surface integrals. Formula di Stokes formula and divergence theorem in 3D (4/2)

Contents

1. Vector spaces and Euclidean spaces: Vector spaces and subspaces. Linear independence. Basis and dimension. Application to linear systems – Euclidean vector spaces. Scalar product, norms and angles. Cauchy-Schwarz inequality. Orthogonality (h Lectures/Exercises 6/4)

2. Matrices and linear sistems: Matrices. Determinant and rank – Linear systems. Theorems of Cramer and Rouché-Capelli. Gauss method – Eigenvalues and eigenvectors. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizzation (8/6)

3. Analytic geometry of the space: Review of analytic geometry of the pane – Cartesian coordinates in the space. Equation of planes and lines. Vector product. Parallelism and ortogonality conditions (2/4)

4. Sequences and series of functions: Review of numerical series – Sequences of functions. Pointwise and uniform convergence. Continuity of uniform limit. Interchanging limit with integral and derivative sign – Series of functions. Uniform and total convergence. Integration by series. Derivative term by term – Power series. Convergence interval and radius. Analiticity. Taylor series (5/3)

5. Functions of more variables: Basic topology in IRn. Limits and continuity – Partial derivatives. Higher order derivatives. Schwarz Theorem – Gradient. Differentiability. Composite functions. Directional derivatives. Level curves. Functions with zero gradient in a convex domain – Taylor formula and second order differentiability. Hessian matrix. Quadratic forms. Matrices with definite sign. Local and global maxima and minima – Vector valued functions and Jacobian matrix (5/5)

6. Differential equations: General solution and Cauchy problem – Equivalence between equations of order n and systems of n first order equations – The Cauchy theorem on local existence and uniqueness with consequences. Global existence. Maximal solution – Equations with separable variables – Linear differential equations. Wronskian and linear independence. Structure of the general integral of linear homogeneous and non-homogeneous differential equations with constant coefficients. Bernoulli equations. Higher order linear homogeneous and non-homogeneous differential equations with constant coefficients – Linear systems (6/6)

7. Curves and curvilinear integrals: Regular curves. Direction and tangent vector. Curve orientation – Length of a curve. Rectifiability. Piecewise regular curves. Curvilinear integral of a function (3/2)

8. Differential forms: Vector fields. Work. Conservative fields – Linear differential forms. Curvilinear integral of a differential form. Exact differential forms. Primitives and potential of the associated vector field – Exactness criterion in connected domains. Closed forms. Exactness criterion in star-shaped domains (4/3)

9. Multiple integrals: Double integrals over normal domains. Reduction formulas – Gauss-Green formulas in the plane. Divergence theorem. Stokes formula – Changing variables. Polar coordinates – Triple and higher dimensional integrals (5/7)

10. SURFACE AND SURFACE INTEGRALS: Regular surfaces. Tangent plane and normal vector – Surface area – Oriented surfaces. Boundary – Surface integrals. Formula di Stokes formula and divergence theorem in 3D (4/2)

	Teaching Methods
	THE COURSE CONSISTS OF 48 HOURS OF THEORETIC FRONTAL LECTURES WITH EXAMPLES AND 42 HOURS OF EXCERCISE SESSIONS, IN TOTAL 90 HOURS (9 CREDITS). THE ATTENDANCE IS MANDATORY. THE STUDENT IS REQUIRED TO ATTEND AT LEAST THE 70% OF THE COURSE, USING HIS PERSONAL BADGE.

	Verification of learning
	THE EXAM CONSISTS OF TWO PARTS: A WRITTEN TEST WITH THEORETICAL AND NUMERICAL EXERCISES FOR APPLYING KNOWLEDGE; AN ORAL EXAM WITH CONCEPTUAL AND TECHNICAL QUESTIONS CONCERNING THE CONTENTS OF THE COURSE TO VERIFY KNOWLEDGE AND UNDERSTANDING AS WELL AS COMMUNICATION SKILLS. The final grade, expressed in thirties, is the result of the written and oral assessment.

	Texts
	N. FUSCO, P. MARCELLINI, C. SBORDONE, ANALISI MATEMATICA DUE, LIGUORI EDITORE P.MARCELLINI, C.SBORDONE, Esercitazioni di Matematica, Volume 2, Parte I, LIGUORI EDITORE P.MARCELLINI, C.SBORDONE, Esercitazioni di Matematica, Volume 2, Parte I, LIGUORI EDITORE Dispense del docente ROBERT A. ADAMS, CRISTOPHER ESSEX, CALCULUS, A COMPLETE COURSE, 7TH EDITION, PEARSON. Lecture notes