Math 442: Honors Real Analysis II


Course Information

Professor: Kiril Datchev
Email: kdatchev@purdue.edu
Lectures: Mondays, Wednesdays, and Fridays 9:30 to 10:20 am, in STON 217.
Office hours: Mondays, Wednesdays, and Fridays 10:20 to 11:10 in STON 217 and Tuesdays 11:10 to 12:00 in REC 122, or by appointment.

Textbook: Calculus on Manifolds, by Michael Spivak. We will also use some material from Introduction to Analysis in Several Variables, by Michael E. Taylor, also available as a preprint.

The topics we will cover include differentiation and integration of functions on Euclidean space, including an introduction to differential forms and manifolds, with examples from differential equations, geometry, and physics.

Grading will be based on

  • Almost weekly homework assignments, worth 20% of the total grade,
  • two in-class midterm exams, worth 40% of the total grade,
  • a final exam, from 7 to 9 pm in STON 217 on Friday May 7th, worth 40% of the total grade.
  • If any problem, such as illness, quarantine, or anything else, interferes with your ability to attend class and do the required work, please email me so that we can arrange a suitable accomodation.

    Homework

    Homework is due on paper at the beginning of class on Wednesdays. Here are the assignments and associated handouts:

    Homework 1, due January 27th. See also this handout from Rudin on length of a curve, and also notes on geodesics.
    Homework 2, due February 3rd.
    Homework 3, due February 10th. See also notes on interchanging derivatives and inverse and implicit functions.
    There will be no homework due February 17th, but the first midterm will be Monday February 15th.
    Homework 4, due February 24th. See also notes on ordinary differential equations.
    Homework 5, due March 3rd.
    Homework 6, due March 10th. See also notes on geodesics and this handout from Shifrin's Multivariable mathematics on manifolds in Rn.
    Homework 7, due March 17th. See also notes on geodesics.
    Homework 8, due March 24th. See also notes on geodesics.
    There is no homework due March 31st or April 7th, but the second midterm is on Monday, April 5th. See also notes on partitions of unity.
    Homework 9, due April 14th.
    Homework 10, due April 21st. See also notes on differential forms.
    There is no homework due April 28th, but the final is on Friday, May 7th.


    Additional Resources

    Below are some books recommended for further reading.
    Analysis on Manifolds, by James R. Munkres, covers the material of Spivak in much more detail and depth.

    Three good introductions to partial differential equations, in increasing order of difficulty, are
    Introduction to Partial Differential Equations, by David Borthwick,
    Partial Differential Equations: An Accessible Route through Theory and Applications, by András Vasy,
    Partial Differential Equations, by Lawrence C. Evans.


    Finally, a list of general policies and procedures can be found here.