## Mathematics 430## Calculus on Manifolds |

- extend the basic concepts of single-variable calculus to the contexts of multivariable calculus and calculus on manifolds.
- develop and demonstrate a level of expertise in mathematical reasoning appropriate to a challenging upper-level mathematics course.

These additional references relate well to this course.

- MIT OpenCourseWare hosts the very similar MIT course 18.101 Analysis II, Fall 2004 by Victor Guillemin.
- MIT OpenCourseWare also hosts a course in differential geometry based on a highly regarded text by Manfredo do Carmo, 18.950 Differential Geometry, Spring 2005. Readings for that course include a handout on the inverse and implicit function theorems.
- John Lee's course materials for Math 310, Introduction to Mathematical Reasoning, at the University of Washington includes helpful advice on writing mathematics and on assessing the mathematical writing of others.

The link (pdf) is to a file which was generated from an Excel spreadsheet. It can be displayed by your web browser or by an application such as Adobe Acrobat or Preview.

- Math 430 TTh Spring 2006 (pdf)

Much of our course will center on working through a substantial set of exercises taken from the Munkres text and from the online MIT OpenCourseWare materials written by Victor Guillemin which complement that text. Developing clear, clean, and elegant solutions to those problems, and explaining those solutions to each other, will take up much of our class time. You are then to write up your own versions of those solutions and hand them in as homework. The class discussions represent a community effort and may not extend to all of the exercises for which you are responsible. The homework for this course consists in writing a representation of your own understanding of the solutions to these exercises, whether discussed in class or not. Although the class sessions are collaborative and explorative, writing the homework solutions is an individual effort and is definitive. The goal is to engage and internalize the central mathematical issues of correctness, precision of expression, elegance, and style.

- Exercises for Math 430 TTh Spring 2006 (pdf)

- Chapter 1: The Algebra and Topology of R
^{n} - Chapter 2: Differentiation
- Chapter 3: Integration
- Chapter 4: Change of Variables
- Chapter 5: Manifolds
- Chapter 6: Differential Forms
- Chapter 7: Stokes Theorem
- Chapter 8: Closed and Exact Forms

Sewanee's tradition of cordial and constructive student-faculty relationships is one of its great strengths, and I am very happy to support that tradition. If you would like to talk with me, please make an appointment, either when you see me in class or in the hallways or in my office (WL120), or by email (cparrish@sewanee.edu), or by voice mail message (x1333).

Homework is due on Thursdays!

- 3 review examinations 60 %
- homework and such 20 %
- final examination 20 %