MATHS 471: Real Analysis 1 (3)

Syllabus

1.      Prerequisite:  MATHS 215, 267, and 371, or permission of department chair.

Note: Students with strong performance in MATHS 166, 215, and 267 may request permission to substitute MATHS 471 for MATHS 371.

2.  Course Description: Properties of the real numbers.  Cardinality.  Topological properties of metric spaces: compactness, completeness, connectedness. Sequences and series.  Continuous functions. Differential calculus of real- and vector-valued functions of one real variable.

3.  Course Objectives: The main objective of the course is a careful and rigorous treatment of the main ideas of differential calculus, as first encountered by the student in a freshman-level calculus course, and expanded upon in MATHS 371. The concepts of sequences in metric spaces and numerical series will also be studied. MATHS 471 includes a basic treatment of metric spaces, providing the necessary technical language and framework for the discussion of modern analysis topics.  Students completing the course will gain a deeper understanding of differential calculus and they will gain experience in writing proofs and solving problems related to the content of the course. 

4.  Course Rationale: This course primarily serves Mathematics majors and graduate students (since the course is taught with MATHS 571).  Students who take this course will have a better understanding of the theoretical underpinnings of differential calculus.  Sequences and infinite series will be treated in depth. Since the course provides essential knowledge for further study in analysis, topology, geometry, and applied mathematics, it is an essential course for students contemplating earning a doctorate in the mathematical sciences. 

5.  Course Content: Properties of the real numbers, including  completeness and least upper bound property; cardinality of sets; theory of metric spaces, with emphasis on Euclidean space, including  open and closed sets, Cauchy and convergent sequences, compactness, completeness, and connectedness; numerical sequences and infinite series, including convergence tests, rearrangement, and multiplication of series; continuous functions between metric spaces, including the Extreme Value Theorem, Intermediate Value Theorem, and uniform continuity; differentiability of functions of one variable, including the  product rule, quotient rule, chain rule, Mean Value Theorem, continuity of derivatives, l’Hospital’s Rule, and Taylor’s theorem. 

6.  Course Format: lecture/discussion.  The amount of material to be covered does not allow for complete proofs to be given in class of all the results which are important. The instructor is responsible for choosing a representative sample of proofs that illustrate the most common techniques.  Students are expected to understand other proofs through careful reading of the text.  

7.  Methods of Evaluating Student Performance:  Typical components are homework assignments, projects, take-home exams (possibly involving group work), in-class exams, student presentations, and oral exams.  The evaluation and weight of these components are at the discretion of the instructor.  Students may be evaluated on their understanding of the content material of the course, their ability to solve problems related to the course material, and on their ability to communicate mathematically to others orally and/or in writing.  

8.  Evaluation of the Course: The instruction of the course is evaluated by departmental student evaluations and peer evaluation.  The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.

Ahmed Mohammed, Michael Karls, Spring 2007