MATH 312 – Real Analysis 2

Certainly, here’s a sample syllabus for a Real Analysis 2 course, which typically covers more advanced topics in real analysis. Please note that this is a general example, and you should adapt it to meet the specific requirements and standards of your educational institution:

Syllabus

Course Title: Real Analysis 2Course Description: Real Analysis 2 is the second part of a two-semester course that delves deeper into the theory of real analysis. This course covers topics such as integration theory, sequences and series of functions, and selected advanced topics in real analysis.Course Objectives: By the end of this course, students should be able to:Understand advanced concepts in real analysis, including Lebesgue integration.Analyze and apply the theory of Lebesgue integration to solve mathematical problems.Study and prove properties of sequences and series of functions.Gain proficiency in mathematical writing and proof techniques.Apply real analysis concepts to more complex mathematical problems and real-world applications.

Textbook:
  • [Insert Real Analysis 2 Textbook Title and Author(s)]
  • Materials:
  • Notebook or binder for class notes and assignments.Graphing calculator (if required by the school or teacher).Pencils, erasers, and a ruler.
  • Grading:
  • Homework and Class Participation: XX%Quizzes and Assignments: XX%Midterm Examinations: XX%Final Examination: XX%
  • Course Outline:Unit 1: Lebesgue Integration
  • Introduction to Lebesgue measure and integration.Lebesgue measurable sets and functions.The Lebesgue integral and its properties.Convergence theorems: dominated convergence and monotone convergence.
  • Unit 2: Sequences and Series of Functions
  • Pointwise and uniform convergence of sequences of functions.Series of functions and power series.Weierstrass Approximation Theorem.
  • Unit 3: Advanced Topics in Real Analysis
  • Lp spaces and their properties.Differentiation of functions of several variables.Implicit and inverse function theorems.Introduction to functional analysis (if time permits).
  • Unit 4: Review and Final Exam PreparationNote: This is a general example of a syllabus for an advanced Real Analysis 2 course. It’s essential to adapt it to meet the specific needs and standards of your educational institution. The percentages for grading, the textbook, and materials may vary. Additionally, consult with your institution’s curriculum guidelines and any state or district standards that may apply. Real Analysis 2 is typically a college-level course, and the level of rigor and depth may vary based on the institution’s curriculum.

    Free Textbooks


    Finding a completely free textbook for an advanced course like Real Analysis 2 can be challenging, as these textbooks often involve specialized topics and in-depth mathematical content. However, I can recommend some open educational resources (OER) and freely available online textbooks that cover advanced real analysis topics. Here are a few options:

    1. “Real Analysis” by Terry Tao:
      • This free online textbook by Terry Tao covers a wide range of topics in real analysis, including Lebesgue integration and advanced analysis concepts. It’s suitable for advanced students.
      • Website: https://terrytao.wordpress.com/analysis-i/
    2. “Real and Complex Analysis” by Walter Rudin:
      • While not free, “Real and Complex Analysis” by Walter Rudin is a classic textbook in real analysis. Some universities provide free access to this book through their libraries or websites.
    3. “Measure Theory and Integration” by Michael E. Taylor:
    4. MIT OpenCourseWare (OCW):
      • MIT OCW provides free access to course materials from Massachusetts Institute of Technology (MIT) courses, including those related to real analysis. You can find lecture notes, assignments, and exams related to real analysis.
      • Website: https://ocw.mit.edu/index.htm
    5. Harvard Online Learning – Real Analysis Course:
    6. “Principles of Mathematical Analysis” by Walter Rudin:
      • While not free, “Principles of Mathematical Analysis” by Walter Rudin is a classic textbook that covers a wide range of real analysis topics.

    Remember to check the licensing terms and any copyright restrictions when using online resources. Additionally, inquire with your institution’s library or online resources to see if they provide access to advanced mathematics textbooks and materials.

    Free MOOCs

    As of my last knowledge update in January 2022, finding a completely free Massive Open Online Course (MOOC) specifically focused on Real Analysis 2 might be challenging due to the advanced nature of the subject. However, you can explore MOOC platforms for related courses in advanced mathematics and analysis, as some of them may offer free auditing options. These courses may cover topics that align with the curriculum of a Real Analysis 2 course. Keep in mind that while auditing is often free, fees may apply if you want certificates or additional features. Here are some platforms where you can explore relevant courses:

    1. Coursera:
      • Coursera offers advanced mathematics courses from universities and institutions. While auditing is often free, fees may apply if you want certificates or additional features.
      • Website: https://www.coursera.org/
    2. edX:
      • edX provides courses in advanced mathematics, including topics related to analysis. You can audit these courses for free, but certificates may have associated fees.
      • Website: https://www.edx.org/
    3. MIT OpenCourseWare (OCW):
      • MIT OCW offers free access to course materials from Massachusetts Institute of Technology (MIT) courses, including those related to advanced mathematics and analysis.
      • Website: https://ocw.mit.edu/index.htm
    4. Khan Academy:
      • Khan Academy offers a wide range of free math courses and tutorials that cover advanced topics, including calculus and mathematical analysis.
      • Website: https://www.khanacademy.org/
    5. Stanford Online:
      • Stanford University offers some free online courses in advanced mathematics and related fields. Check their website for current offerings.
      • Website: https://online.stanford.edu/courses
    6. Harvard Online Learning:

    While these courses may not be specifically tailored to Real Analysis 2, they cover advanced mathematical topics that are relevant to the subject. Be sure to explore the course descriptions and offerings on each platform to find courses that align with your learning objectives. Additionally, verify the availability of free auditing options for the courses you are interested in.

    Leave a Comment

    This site uses Akismet to reduce spam. Learn how your comment data is processed.