MATH Honors 554 and 703-I Real Analysis I  - Fall 2008 Professor Robert Sharpley Meets: TTh  2:00 - 3:30 in LeConte 121 Instructor Information  Office Hours:   TTh 3:30-4:15  (LC 429)  Additonal Hours:   Wed: 10:00-11:45 (Sum 206)

Course Information

Description:   This Honors course is a rigorous treatment of analysis required for a fuller understanding of the calculus, as well as preparation for graduate school in mathematics and other disciplines requiring analytical and numerical solution of equations arising from mathematical modeling. Standard topics of an introductory analysis course will be covered, along with additional concepts which do not necessarily follow the text. Attendence is required and the exams will be over the lectures and homework. Course topics include:

• Countable and uncountable sets, the real numbers, order, least upper bounds, and the Archimedean property.
• Metric spaces: topology, open and closed sets, convergent sequences, completeness, compactness and the Heine-Borel Theorem for the real line, connectedness.
• Limits; Continuous functions and their properties, limits, continuous functions on a compact metric space, continuous functions on a connected metric space; intermediate and extreme value theorems, uniform continuity, monotone functions and inverses.
• Derivatives and their properties, the chain rule, Rolle's theorem and the Mean Value theorem, Taylor's theorem, L'Hospital's rule.
• The Riemann integral, its properties, and the Fundamental Theorem of Calculus.

Recommended Text:  Principles of Mathematical Analysis, (3-rd edition) by Walter Rudin, McGraw-Hill Science/Engineering/Math, 1976.
      [ISBN 978-0070542358 (also available in paperback)]

Grading: Three tests (indicated on the calendar below) will each factor as 20% of the course grade. The homework, turned in on a regular basis, counts 15%, with the comprehensive final exam accounting for the remaining 25%. Make-up tests are not normally given. Graduate Students will be required to complete additional problems for homework and on exams, and complete a term paper.

Expected Outcomes: This course is considered as an undergraduate 'Finishing Course' for Mathematics majors. As such, students will be expected to master the concepts of analysis (in particular the listed topics), provide mathematical rigor to arguments, and demonstrate the ability to effectively write mathematical prose.

Attendance: Classroom attendance is required according to official university 10% policy. Make-up tests are not normally given for missed examinations.

Important Course Dates:

Aug 21   (Thur) Aug 27   (Wed) Sept 1   (Mon) Sept 18   (Thur) Oct 2   (Thur) Oct 9-10   (Thur-Fri) Oct 28   (Tues) Nov 4   (Tue) Nov 25, Takehome Exam due   (Tue) Nov 26-30   (Wed.-Sun.) Dec 5   (Fri) Classes Begin Last Day for Withdrawal w/o penalty Labor Day Holiday (no classes) Test 1 Last Day for Withdrawal w/o WF grade Fall Break - no classes Test 2 (new date!) General Election (no classes) Test 3 Thanksgiving Holidays (no classes) Classses End Dec 6   (Sat) Dec 9   (2 pm, Tue) Reading Day Final Exam

Prerequisites: Admittance is restricted to students in the Honors College and to students approved through special petition to the Director of Undergraduate Studies, Dr. Douglas Meade.

  Lectures Notes

  Tests and Solutions

  Homework Assignments

  Link to Interactive Real Analysis

This page maintained by Robert Sharpley (sharpley@math.sc.edu) and last updated September 24, 2008.  This page ©2008-2009, The Board of Trustees of the University of South Carolina.  URL: http://www.math.sc.edu/~sharpley/math554_08