# Math 442: Honors Real Analysis II

## Course Information

Professor: Kiril Datchev

Email: kdatchev@purdue.edu

Lectures: Tuesdays and Thursdays, 1:30 to 2:45, in SCHM 309.

Office hours: After class, Wednesdays 1-3 in MATH 602, or by appointment.

Textbooks: We will draw material from *
Introduction to Analysis in Several Variables*, by Michael E. Taylor, also available as a preprint, from *Multivariable Mathematics*, by Theodore Shifrin, and from *Differential Forms and Applications*, by Manfredo P. do Carmo.

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 35% of the total grade,
an in-class midterm exam, worth 25% of the total grade,
a final exam, as scheduled here, worth 40% of the total grade.

### Homework

Homework is due on paper in class on Thursdays. Here are the assignments:

Homework 1, due January 18th.

Homework 2, due January 25th.

Homework 3, due February 1st.

Homework 4, due February 8th.

Homework 5, due February 15th.
See also these notes on geodesics.

Review problems for the midterm on February 27th.

Homework 6, due March 7th. See also these notes on ODEs.

Homework 7, due March 21st. Notes on iterated integrals.

Homework 8, due March 28th, is 2.2.7(1) and (2), 2.3.5(1), 2.3.12, and 2.3.13(4), from the notes on iterated integrals.

Homework 9 due April 4th.

Homework 10, due April 11th, is use Proposition 2 from Section 1 of do Carmo's book to do Exercises 4 and 7 there, and read page 7 of Sadun's notes and do Exercises 5 and 6, using 5 to help with 6.

Homework 11, due April 18th, is Exercises 4.1.4 and 4.2.4 from the notes on iterated integrals.

Review problems for the final on April 30th.

### Additional Reading

How much detail?

The two classic introductions to multivariable analysis and differential forms are *Calculus on Manifolds*, by Michael Spivak and *Analysis on Manifolds*, by James R. Munkres. The former is concise and all business while the latter covers the material in more detail and depth.

A more unconventional but beautiful and insightful introduction to forms can be found in Act V of Tristan Needham's Visual Differential Geometry and Forms.

Another good source on forms is Lorenzo Sadun's notes.

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