Math312

Course Description form

Faculties and departments requiring this course (if any): Faculty of Science-

     Mathematics Department

Objectives:  Prefer In points

1- to make the student acquainted with some classical and fundamental techniques in mathematical analysis.

2- to give to the student some justifications and extensions of concepts he used to manipulate without theoretical background.

3- to make the student familiar with the manipulation of the concept of the limit process.

Contents: Prefer In points

1- Part I. The Riemann integral: The construction with Darboux sums. Criteria of Integrability. Application to continuous functions and monotonic functions on a compact interval. The fundamental theorem of calculus.

2- Part II. Sequences and series of functions: Simple convergence and uniform convergence. Criteria for uniform convergence. Continuity, integrability and differentiability in the limit under uniform convergence. (This part should start by a brief review on numerical series).

3- Part III. Metric spaces: Metric on a set. Basic metrics on the n-Euclidian space. Metric space. Open balls, closed balls, spheres. Subspaces. Topology on a metric space. Open sets. Neighborhoods. Convergent and Cauchy sequences in a metric space. Continuous functions. Completeness. Compact sets in a metric space.

4- Part IV. Function of several variables. Differentiation and partial differentiation. Functions of two variables. Local behavior: The implicit function theorem and the inverse function theorem.

Course Outcomes:

A. Knowledge:

    (Specific facts and knowledge of concepts, theories, formula etc.)

    The goal is that student becomes fairly able to master basic techniques of Real Analysis, dominated by the concept of the limit: limit in Euclidian spaces and more generally in metric spaces. Local behavior of functions of several variables. 

B-Cognitive Skills:

    (Thinking, problem solving )

    The students have to master the techniques of proof of the main theorems in each part of the course. This will improve their thinking, so they will be able to solve the standard problems of the course by taking personal initiative.

C- Interpersonal skills and responsibilities:

     (group participation, leadership, personal responsibility ,  ethic and moral behavior, capacity for  self directed learning)

    All teaching strategies should basically consist in developing communication and increasing the opportunities for the student to express oneself. This leads to favor working in group sessions guided by the teacher and the student must be an active partnership during the meeting of the sessions.   

D- Analysis and communication:

        (communication, mathematical and IT skills)

    Ability to seek information about the subject considered, or some complement of it, from text books proposed by the teacher. The student must learn how to read and understand a topic in a book and definitely give up the memorizing reflex.

     Assessment methods for the above elements: Class discussions on the problems given to the groups of students.

     Methods of assessment of students include:

     (a) Cross questionings and class discussions;

     (b) Simple unseen problems proposed in Quizzes;

     (c) Home assignments, midterm exams,  and final exam.

Text book:

          Principles of Mathematical Analysis, By Walter Rudin, Wiley.