"GH. ASACHI" TECHNICAL UNIVERSITY OF IASI
FACULTY: MECHANICAL ENGINEERING
SPECIALIZATION: ALL PROFILES
SUBJECT NAME: MATHEMATIC ANALYSIS (CALCULUS)
ACADEMIC YEAR: 2006/2007
Professorial Council of Mechanical Engineering
PROGRAMME OF MATHEMATIC ANALYSIS (CALCULUS)
1. Full Professor: Ion Al. Craciun
2. Specialization of the Professor: PhD in Mathematics, Elasticity
3. Type of matter: fundamental and compulsory education
4. The structure in an eventually educational plane
Semester
I
Hours/Week
Hours/Semester
Course
Seminar
Course
Seminar
2
2
28
28
Form of
Verification
Credits
Examination
5
5. The Objectives of the discipline
As always, the fundamental objectives of this discipline are to teach the
mathematics of calculus and to provide the training students for the use calculus effectively
in their later academic and professional work. To accomplish this, we preserve accessible
the lecture’s mathematical level, its orientation toward applications and its concentration
on worked examples A number of exercises is used to show the connection between
calculus and some of the numerical methods used in other courses.
Calculus was invented to solve problems in physics, and astronomy, and although it
has since developed into a far-reaching mathematical discipline in its own right, most of its
applications outside of mathematics still involve science and engineering. The applications
of this discipline are directed mainly along these lines. Typical applications include
calculating of extreme values, center of mass, moments, work, describing fluid flow etc.
However, in recent years, the calculus has become important in many other fields,
including economics, business, the life sciences, and even the physics of sports. Whenever
we feel we make the connections between calculus and real life.
6. The methodology of giving the lectures
Our aim in giving the lectures and applications of this discipline is to present the
fundamentals of calculus in the clearest possible way. Pedagogy is the main consideration;
formalism is secondary. Where possible, basic ideas are studied by means of computational
examples and geometrical interpretation.
Our treatment of proofs varies. Those proofs that are elementary and have significant
pedagogical content are presented precisely, in a style tailored for beginners. A few proofs
that are more difficult, but pedagogically valuables, are considered optional. Still other
proofs are omitted completely, with emphasis placed on applying the theorem. Whenever a
proof is omitted, we try to motivate the result, often with a discussion about its
interpretation in 2-space or 3-space.
It is a pedagogical axiom that a teacher should proceed from the familiar to the
unfamiliar and from the concrete to the abstract. The ordering of the chapters reflects our
adherence to this tenet.
7. CONTENTS OF LECTURES
Chapter 1 – The metric and the vector space IRn
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
Definition of the set IRn and the Metric Space IRn
The Vector Space IRn
The Banach Space IRn
The Inner Product (Euclidian) Space IRn
The Hilbert Space IRn
The Topological Space IRn
Direction in the Vector Space IRn
Chapter 2 – Sequences and Series
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
Sequence of points in IRn
Series of vectors in IRn
Numerical Series with Positive Terms: Comparison and Integral Tests
Numerical Series with Positive Terms: Ratio and Root Tests
Absolute Convergence
Alternating Series and the Leibniz’s Test
Function Sequences
Function Series
Power Series: Convergence; Derivability; Integration.
Chapter 3 – Real Functions of Two and More Real Variables
3.1. Real Functions of Two and More Real Variables
3.2. Directional Derivative
3.3. Partial Derivability of First Order and Partial Derivatives of First Order
3.4. The Chain Rule
3.5. Partial Derivatives of Higher Order
3.6. The Schwarz and Young Tests of Equality of Second Order Derivatives
3.7. The Gradient and the Hamilton Operator
3.8. First Order Differentiability and First Order Differential
3.9. Linear Approximation and Error Estimates
3.10. High Order Differentials
3.11. The First Order Differential and the Partial Derivatives of a Composite Real Function
3.11. High Order Differentials and Partial Derivatives of a Composite Real Function
3.12. Taylor Formula of a Real Function with a Real Variable
3.13. Taylor Formula of a Real Function with Two and More Real Variables.
Chapter 4 – Applications of Differential Calculus
4.1.
4.2.
4.3.
4.4.
4.5.
Maxima, Minima and Saddle Points of a Real Function of Two and More Variables
Necessary Condition of Extreme and Fermat’s Theorem
Quadratic Forms on IRn and their Classification
The Second Derivatives Test
Real Functions and Systems of Real Functions of One and More Variables Defined
Implicit
4.6. Maxima and Minima of Functions Defined Implicit
4.7. Constrained Maxima and Minima. The Method of Lagrange Multipliers
4.8. Extreme of Real Functions Defined on Compact Sets
4.9. Functional Dependence
4.10. Regular Transformations
4.11. Change of Variables in Differential Expressions.
Chapter 5 – Extending of Definite Integral
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
5.9.
Definition and Classification of Improper Integrals
Examples of Improper Integrals
Leibniz – Newton Formula
Reducing Improper Integrals to Numerical Sequences and Numerical Series
Extending Methods of Evaluating Integrals to the Case of Improper Integrals
Absolute Convergence of an Improper Integral and Convergence Tests
Proper Integrals Dependent on Parameter
Properties of Improper Integrals Dependent of Parameter
Simplest Improper Integrals Dependent of Parameter and their Continuity,
Derivability and Integration
5.10. Uniform Convergence of an Improper Integral Dependent of Parameter
5.11. Tests of Uniform Convergence of an Improper Integral Dependent of Parameter.
Chapter 6 – Line Integrals
6.1.
6.2.
6.3.
6.4.
6.5.
6.6.
6.7.
6.8.
6.9.
Paths and Curves
Definition of the First Type Line Integral
Evaluating of the First Type Line Integrals
Properties of the First Type Line Integrals
Applications in Geometry and Engineering of the First Type Line Integrals
Definition of the Second Type Line Integrals
Evaluating of Second Type Line Integrals
Dependence of Path of a Second Type Line Integral
Reconstructing of a Function from Its Total Differential.
Chapter 7 – The Double Integral
7.1. Definition and Basic Properties of Double Integral
7.2.
7.3.
7.4.
7.5.
7.6.
Evaluating of Double Integrals
Change of Variables in a Double Integral
Curvilinear Coordinates
Applications in Geometry of Double Integrals: Evaluating Volume; Computing Areas
Applications in Mechanics of a Double Integral: Mass of a Plate; Coordinates of the
Center of Gravity of a Plate; Moments of Inertia of a Plate; Luminous Flux on a Plate;
Flux of a Fluid Through the Cross Section of a Channel
Chapter 8 – Triple Integrals
8.1. Definition and Basic Properties of a Triple Integral
8.2. Evaluating of a Triple Integral
8.3. Some Applications of Triple Integrals in Engineering and Geometry: Computing
Volumes; Finding the Mass of a Solid from its Density; Moments of Inertia;
Determining the Coordinates of the Center of Gravity; Gravitational Attraction of a
Material Point by a Solid
8.4. Change of Variables in Triple Integrals
8.5. Triple Integrals in Curvilinear Coordinates.
Chapter 9 – Surface Integrals
9.1.
9.2.
9.3.
9.4.
9.5.
9.6.
Definition of the First Type Surface Integral
Reducing Surface Integral to Double Integral
Some Applications of Surface Integral in Mechanics
One Sided and Two-Sided Surfaces
Definition of Surface Integral of the Second Type
Reducing Surface Integral of the Second Type to Double Integral.
Chapter 10 – Integral Formulae
10.1.
10.2.
10.3.
10.4.
Hamilton Operator Applied to Vector Fields
Derivation of Stokes Formula and Applications
Green’s Theorem in the Plane
The Divergence Theorem.
Chapter 11 – Differential Equations
11.1. Introduction in Differential Equations Theory
11.2. General Notions on First Order Differential Equations of First Degree. Picard‘s
Existence and Uniqueness Theorem
11.3. First Order Differential Equations with Separate Variables
11.4. Homogeneous Differential Equations of First Order and Differential Equations
Reducible to First Order Homogeneous Differential Equations
11.5. Exact Differential Equations of First Order
11.6. Linear Differential Equations of First Order
11.7. Differential Equations of First Order Reducible to Linear First Order
Differential Equations: Bernoulli Equation; Riccati Equation
11.8. High Order Differential Equations Reducible to First Order Differential Equations.
Selected References
[1]. BUDAK, B. M., FOMIN, S. V. – Multiple Integrals, Field Theory and Series. An
Advanced Course in Higher Mathematics. Mir Publishers, Moscou – 1973
[2]. CALISTRU, N., CIOBANU, Gh. – Curs de analiza matematica. Vol. I. Institutul
Politehnic Iasi, Rotaprint, Iasi – 1988
[3]. CHIRITA, S. – Probleme de matematici superioare. Editura Academiei Romane,
Bucuresti – 1989
[4]. CRACIUN, I. Al. – Calcul diferential. Editura Lumina, Bucuresti - 1997
[5]. CRACIUN, A., CRACIUN, I. Al., ISPAS, M. – Analiza matematica. Partea I.
Culegere de probleme de calcul diferential. Editura Politehnium,
Iasi – 2004
[6]. CRAIU, M., TANASE, V. V. – Analiza matematica. Editura Didactica si Pedagogica,
Bucuresti – 1980
[7]. IONESCU, D. V. – Ecuatii diferentiale si integrale. Editia a doua. Editura Didactica
si Pedagogica, Bucuresti – 1980
[8]. NICOLESCU, M., DINCULEANU, S., MARCUS, S. – Analiza matematica. Vol. I.
Editia a IV-a. Editura Didactica si Pedagogica, Bucuresti – 1971
[9]. PRECUPANU, A. – Bazele analizei matematice.Editura Universitatii “Al. I. Cuza”,
Iasi - 1993
[10]. ROGAI, E. – Exercitii si probleme de ecuatii diferentiale si integrale. Editura
Tehnica, Bucuresti - 1965.
[11]. ROSCULET, M. – Analiza matematica. Editia a patra. Editura Didactica si
Pedagogica, Bucuresti – 1984
[12]. SMIRNOV, V. – Cours de mathématiques supérieures. Tome I. Éditions Mir,
Moscou - 1972
[13]. STANASILA, O. – Analiza matematica. Editura Didactica si Pedagogica,
Bucuresti – 1981
[14]. SYKORSKI, R. – Advanced Calculus. Functions of several variables. PWN –
Polish Scientific Publishers, Warszawa – 1969.
September 30, 2006
Entitled Professor,
Professor Ion Al. Craciun
Head of Department of Mathematics,
Associate Professor Dr. Constantin Popovici