Math 217 - Honors Advanced Calculus, I

Instructor

Volker Runde

Office hours

TBA

Syllabus

The real number system and finite dimensional Euclidean space

• axiomatic introduction of the real numbers
• the Euclidean space R^N
• functions
• topology in R^N

Limits and continuity

• limits of sequences
• limits of functions
• global properties of continuous functions
• uniform continuity

Differentiation in R^N

• differentiation of real valued functions of a real variable
• partial derivatives
• vector fields
• total differentiability
• Taylor's theorem
• classification of stationary points

Integration in R^N

• content in R^N
• the Riemann integral in R^N
• calculation of integrals
• Fubini's theorem
• integration in polar, spherical, and cylindrical coordinates

Textbooks

None required, but the following are recommended.
1. Robert G. Bartle, The Elements of Real Analysis, Second Edition. Jossey-Bass, 1976.
2. Patrick M. Fitzpatrick, Advanced Calculus. PWS Publishing Company, 1996.
3. James S. Muldowney, Advanced Calculus Lecture Notes for Mathematics 217-317, I, Third Edition.
4. William R. Wade, An Introduction to Analysis, Second Edition. Prentice Hall, 2000.

I will more or less follow my old TeXed lecture notes. I plan to rework these notes as the term progresses and make the revised version also available online.

The mathematics magazine in the Sky is for high school students, but you may find some of the articles interesting:

• How do I love thee? Let me count the ways! (about how to compare the sizes of infinite sets);
• The Banach-Tarski paradox or What mathematics and miracles have in common (about a bewildering consequence of the axiomatic method);
• Induction principle (belongs into everyone's mathematical toolkit).
• Weierstraß (a biographical sketch of the man who introduced and into calculus).

Grading

The grade will be based on weekly homework assignments (30%), an in-class-midterm on October 20 (20%), and a final (50%).