MATHS 2100 - Real Analysis II

North Terrace Campus - Semester 2 - 2014

Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called ``analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable. Topics covered are: Basic set theory. The real numbers, least upper bounds, completeness and its consequences. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions, uniform continuity. Differentiation, the Mean Value Theorem. Sequences and series of functions, pointwise and uniform convergence. Power series and Taylor series. Metric spaces: basic notions generalised from the setting of the real numbers. The space of continuous functions on a compact interval. The Contraction Principle. Picard's Theorem on the existence and uniqueness of solutions of ordinary differential equations.

  • General Course Information
    Course Details
    Course Code MATHS 2100
    Course Real Analysis II
    Coordinating Unit School of Mathematical Sciences
    Term Semester 2
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3.5 hours per week
    Prerequisites MATHS 1012
    Course Description Much of mathematics relies on our ability to be able to solve equations, if not in explicit exact forms, then at least in being able to establish the existence of solutions. To do this requires a knowledge of so-called ``analysis", which in many respects is just Calculus in very general settings. The foundations for this work are commenced in Real Analysis, a course that develops this basic material in a systematic and rigorous manner in the context of real-valued functions of a real variable.

    Topics covered are: Basic set theory. The real numbers, least upper bounds, completeness and its consequences. Sequences: convergence, subsequences, Cauchy sequences. Open, closed, and compact sets of real numbers. Continuous functions, uniform continuity. Differentiation, the Mean Value Theorem. Sequences and series of functions, pointwise and uniform convergence. Power series and Taylor series. Metric spaces: basic notions generalised from the setting of the real numbers. The space of continuous functions on a compact interval. The Contraction Principle. Picard's Theorem on the existence and uniqueness of solutions of ordinary differential equations.
    Course Staff

    Course Coordinator: Associate Professor Nicholas Buchdahl

    Course Timetable

    The full timetable of all activities for this course can be accessed from Course Planner.

  • Learning Outcomes
    Course Learning Outcomes
    On successful completion of this course, students will be able to

    1. describe fundamental properties of the real numbers that lead to the formal development of real analysis;

    2. comprehend rigorous arguments developing the theory underpinning real analysis;

    3. demonstrate an understanding of limits and how they are used in sequences, series, differentiation and integration;

    4. present an overview of the basic properties of metric spaces;

    5. construct rigorous mathematical proofs of basic results in real analysis;

    6. appreciate how abstract ideas and rigorous methods in mathematical analysis can be applied to important practical problems.

    University Graduate Attributes

    This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

    University Graduate Attribute Course Learning Outcome(s)
    Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. all
    The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 1,3,4
    An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 2,5,6
    Skills of a high order in interpersonal understanding, teamwork and communication. 5
    A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 6
  • Learning Resources
    Required Resources
    None.
    Recommended Resources
    There are many books on real analysis available in the library. Amongst those, the following is a selection of some that are very compatible with the level and objectives of this course:
    1. Belding & Mitchell: "Foundations of Analysis";
    2. Fitzpatrick: "Real Analysis";
    3. Gaughan: "Introduction to Analysis";
    4. Lárusson: "Lectures on Real Analysis".
    Online Learning
    Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.

    "Elementary Real Analysis" by Thomson, Bruckner and Bruckner is available for free electronic download from this site.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    This course relies on lectures as the primary delivery mechanism for the material. Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding.
    Workload

    The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

    Activity Quantity Workload hours
    Lecture 36 90
    Tutorials 6 18
    Assignments 6 42
    Test 1 8
    TOTALS 158
    Learning Activities Summary
    Lecture Outline

    1. Basic set theory. The real numbers and their defining properties. (Lectures 1-3)
    2. Sequences: convergence, properties of limits, subsequences, Cauchy sequences. (Lectures 4-6)
    3. Limits superior and inferior. The topology of the real numbers. (Lectures 7-9)
    4. Continuity. The key properties of continuous real-valued functions of a real variable. (Lectures 10-12)
    5. The Riemann integral. (Lectures 13-14)
    6. Differentiation. Mean Value theorems, l'Hôpital's rules. (Lectures 16-18)
    7. The inverse function theorem. The fundamental theorem of calculus. (Lectures 19-21)
    8. Series. Tests for convergence.  (Lectures 22-24)
    9. Power series. Taylor's theorem. Sequences and series of functions. (Lectures 25-27)
    10. Convergence of sequences and series of functions. (Lectures 28-30)
    11. Metric spaces: basic definitions and properties. (Lectures 31-33)
    12. The contraction mapping theorem and Picard's theorem. (Lectures 33-36)

    Tutorials

    Tutorials will be held in each even week, covering material from the preceding two weeks of lectures. Students will be encouraged to play an active role in tutorials, but will not be assessed on this. Attendance at tutorials will comprise 3% of the final mark.
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

    1. Assessment must encourage and reinforce learning.
    2. Assessment must enable robust and fair judgements about student performance.
    3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
    4. Assessment must maintain academic standards.

    Assessment Summary
    Component Task type Due Weighting Objective Assessed
    Tutorial attendance Formative Even weeks 3 All
    Assignments Formative and summative Odd weeks 21 All
    Test Summative Lecture 15 20 1,2,3,5
    Exam Summative Exam period 56 All
    Assessment Related Requirements
    An aggregate score of at least 50% is required to pass the course.
    Assessment Detail
    With the exception of the first assignment, assignments will be distributed in the middle of each even week and due at the end of each odd week.  Each will mainly cover material from the previous weeks' lectures, but may include material from the next one or two lectures.

    The first assignment will be distributed at the start of the first week and due at the end of that week. It will largely cover material from Mathematics I for the purpose of refreshing the key ideas from a first course in Calculus.

    The first assignment will be worth 1% of the final mark.  The remaining five assignments will each be worth 4% of the final mark.

    The midsemester test will be held during the 15th lecture period. It will cover the material covered in lectures up to the end of Lecture 12.  It will be worth 20% of the final mark.
    Submission

    All written assignments are to be submitted to the designated hand-in boxes within the School of Mathematical Sciences.

    Late assignments will not be accepted.

    Assignments will have a two week turn-around time for feedback to students.

    Course Grading

    Grades for your performance in this course will be awarded in accordance with the following scheme:

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    FNS   Fail No Submission
    F 1-49 Fail
    P 50-64 Pass
    C 65-74 Credit
    D 75-84 Distinction
    HD 85-100 High Distinction
    CN   Continuing
    NFE   No Formal Examination
    RP   Result Pending

    Further details of the grades/results can be obtained from Examinations.

    Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

    Final results for this course will be made available through Access Adelaide.

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