Mechanical Engineering

MICOL AMAR
MATTEO BOTTACCHIARI

GENERAL OBJECTIVE

Aim of this course is to learn the basic ideas and techniques of differential and integral calculus in one variable, including numerical sequences and series, ordinary differential equations and complex numbers. With a practical approach, the students can develop those basic skills that are fundamental for the comprehension of more advanced courses in Mathematics, Physics and Engineereing . The objective is pursued by means of classical frontal lessons where the students are encouraged to an active attendance.

SPECIFIC OBJECTIVE

1) Knowledge and understanding: To know the basic ideas of Mathematical analysis in one real variable, with emphasis on logical reasoning, on text comprehension, and to the achievement of those skills necessary in order to solve concrete problems.

2) Applying knowledge and understanding: To use the learned tools to solve problems in Mathematical Analysis and discuss concrete examples; to develop those skills that are necessary in order to apply Mathematical Analysis to the solution of scientific problems, in particular in mechanics.

3) Making judgement: To decide the most appropriate approach to solve a specific problem; to classify those mathematical problems usually faced in pure and applied science.

4) Communication skill: To learn to describe the solution of a mathematical problem, pointing which thecniques can be used, justifying the intermediate steps and uderlining the logical reasonings.

5) Learning skill: To develop the necessary skills to learn Mathematical Analysis with the objective that the student can face most advanced courses.

At the end of this course the student is expected to have learned the fundamental ideas and techniques of differential and integral calculus for functions of one variable, through a practical approach, but without neglecting logical reasoning.
It is essential to acquire skills aimed at solving concrete problems.
The student must achieve the necessary skills to understand the following courses in mathematics, physics and engineering.

In order to understand the lessons and to follow the class it is necessary to have
a good knowledge of the following topics:

Powers, polynomials and their properties.
(Systems of) equations and inequalities of algebraic and irrational type.
Logarithmic properties: equations and inequalities of exponential and logarithmic type.
Absolute value properties and applications to equations and inequalities.
Fundaments of trigonometry and analitical geometry.

The main topics of the course will be the following:
Introduction.
Complex numbers.
Numerical sequences and series.
Functions of 1 real variable: limit, continuity, differential calculus.
Integral calculus in 1-d. Improper integrals.
Ordinary differential equations.

REAL AND COMPLEX NUMBERS (1cfu) Introduction. Natural numbers, integers, rationals and real numbers. Modulus. Summations: sum of the geometric progression. Factorial. Complex numbers: algebraic, trigonometric and exponential form; powers, nth roots, polynomial; equations in the complex field.

FUNCTIONS (1cfu) Basic properties (domain, image, graph); functions of one real variable (boundednees, symmetries, monotonicity, periodicity); operations on graphs. Elementary functions (modulus, powers, esponentials, logarithms, trigonometric functions, hyperbolic functions). Composition and inversion. Inverse trigonometric functions (arcsin, arccos, arctan.

LIMITS FOR FUNCTIONS AND SEQUENCES (2cfu) Numerical sequences. Limits and their basic properties: uniqueness. Squeeze theorems in their basic forms and applications. Monotonic sequences. Asymptotic sequences. The fundamental limit of sin. Other fundamental limits. The number e. Limits of functions of a real variable and their properties. Infinitesimal ed infinities. The symbol «o» in asymptotic analysis. Continuous functions: elementary operations. Discontinuous functions. Asymptotes.
The intermediate value theorem, Weierstrass theorem. Monotonicity and continuity.

DIFFERENTIAL CALCULUS (2cfu) Derivatives and their properties: tangent and linear approximation. Derivability and continuity. Elementary derivatives. Rules of computations. Chain rules. Singular points. Characterization of constant functions. Fermat's theorem on stationary points. Lagrange's theorem and its consequences. Higher derivatives and their applications to graphs of functions.L'Hôpital's rule. Taylor's formula.

SERIES (1cfu) From sequences to series. Basic convergence tests.

INTEGRAL CALCULUS (1cfu) Integration of functions of a real variables. Geometric applications. Mean value theorem. The fundamental theorems of calculus. Antiderivatives. Integration of elementary functions. Integration by parts and by substitutions. Integration of some rational and irrational functions.


ORDINARY DIFFERENTIAL EQUATIONS (1cfu) First order differential equations: separation of variables; linear equations. Second order linear differential equations with constant coefficients: homogeneous and non homogeneous case, the method of undetermined coefficients.

1 CFU = 9 hours of frontal lessons

Bramanti - Pagani - Salsa: Analisi matematica I - Zanichelli
Amar -Bersani: Analisi Matematica I: esercizi e richiami di teoria - edizione AMAZON (Codice ASIN: B0BCRXJM2M)


Available material at the web page
https://www.sbai.uniroma1.it/~micol.amar/Meccanica2010.htm

Lessons and exercises conducted at the blackboard.

Theoretical lectures aim to give the main notions of Mathematical Analysis I, through the introduction of the definitions and the theorems concerning the topics characterizing the program. All the arguments are presented with the aid of examples and counterexamples and in some cases the proofs of the theorems are proposed with all the details.

Exercises aim to makes students able to apply the theoretical tools in order to solve independently practical problems. A part of exercises will be carried out by a co-teacher.

Courses attendance is optional and the teacher does not control the presence. However, due to the complexity of the matter, the attendance is highly recommended both to the theoretical and to the practical lectures, since this is a strong support for the individual work.

The exam is mainly written and is divided into two steps. The first one (over a period of half an hour) is made by a test with multiple-choice containing 13 questions of basic analysis; it is necessary to answer correctly to at least 9 questions. The second part (over a period of two hours and half) is composed by 4 standard exercises + 1 theory question. In order to be admitted to the second step the student has to pass the first one. Both parts of the written exam will take place in the same half-day. In the case where a further deepening of preparation is required, an oral examination shall be performed by the student.

In order to pass the exams, students have to know the basic notions and how to solve simple exercises. In order to achieve a high score, students have to show to be sufficiently autonomous, to handle more advanced problems in Analysis.

The evaluation procedure will be carried out with the aim to verify the achievement of the expected goals.
The preliminary test (30 minutes), based on quite simple exercises deals with the specific objectives nr. 1 and 3.
After the positive evaluation of the test and in the same day, the student faces a more structured written examination (2,5 hours). It consists of four practical problems and a theoretical question. It deals with all the specific objectives (nr. 1, 2, 3, 4 and 5).
In order to pass the exam the student must get the grade 18/30; a basic achievement of the skill to solve the standard mathematical problems of a basic one-variable calculus course is required, togheter with the achievement of the specific objectives nr. 2 and 4 at a basic level and of those nr. 1, 3 and 5 at least at an intermediate level.
In order to get the grade 30/30 e lode the student must prove the full achievement of the skill to solve the standard mathematical problems of a basic one-variable calculus course and of the of all the specific objectives (nr. 1, 2, 3, 4 and 5).
The students may take the exam after the end of the course and the calendar is estabilished by the faculty.

State and prove a theorem.
Write a definition.
Give examples and counterexamples.
Exercises on complex numbers, limits and series, differential and integral calculus, differential equations.

See at the link
https://www.sbai.uniroma1.it/~micol.amar/Meccanica2010.htm