MATH 3220-002

Foundations of Analysis II

Course Description

Instructor: Henryk Hecht, JWB 329, hecht@math.utah.edu

Days/time/place: MTWF  11:50 am-12:40 pm in LCB 225.

Office Hours:  to be decided during first class meeting

Textbook: Joseph L. Taylor, Foundations of Analysis, American Mathematical Society, Providence 2012. ISBN 978-0-8218-8984-8

Many additional notes will be posted, when appropriate.

Prerequisites: “C” or better in MATH 3210

General Goals: The main goal of this course is to provide students with a rigorous approach to the theory of several variables calculus. This is the second course of the MATH 3210–3220 sequence of Foundations of Analysis, a sequence designed to develop the mathematical sophistication of students, while giving them a much deeper understanding of calculus and its foundations than can be provided in the standard courses (MATH 1210, 1220, and 2210). The emphasis is on improving the students’ ability to understand and explain concepts in a logical and complete manner and refine their skill at proofs and mathematical arguments. Students who finish both semesters of the sequence should have the mathematical knowledge and sophistication necessary to do well in 4000 and 5000 level mathematics courses.

Bachelor Degree Requirements Met: This course meets the BS Quantitative Intensive (QI) requirement. This course addresses the following Essential Learning Outcomes: Inquiry and Analysis, Critical Thinking, Problem Solving.

Course Description: The course begins with the definition of topology in Euclidean space. Compactness and connectedness are introduced in this context, and the Heine–Borel theorem is one of the main theorems students prove in this part of the course. The definitions of limits, continuity and convergence are revised from the topological point of view. The course moves on to give a rigorous approach to differentiation in several variables, and Taylor formula. This part includes keystone theorems such as the Inverse Function Theorem, the Implicit Function Theorem, Fubini’s Theorem, and the Change-of-Variable Formula, and applications to optimization via Lagrange multipliers, parameterization of higher dimensional surfaces in Euclidean spaces, and computation of their tangent spaces and volumes. The course covers most or all of the following chapters from the textbook:

• Chapter 7: Convergence in Euclidean Spaces
• Chapter 8: Functions on Euclidean Spaces
• Chapter 9: Differentiation in Several Variables
• Chapter 10: Integration in Several Variables

Course Structure: This course is mainly lecture based, with the instructor presenting material at the blackboard. However, students are strongly encouraged to ask questions, and become engaged in discussions.

Grading: Grading is based on homework, mid-term exams, and a final comprehensive exam. Homework is assigned on a weekly basis; many of the homework exercises involve proving theorems or providing examples that illustrate the course material. Grading is based on the following evaluation method:

• Weekly homework assignments, counting 30% toward the final
• Two in-class midterms, each counting 20% toward the final
• Final exam, counting 30% toward the final

The lowest three homework grades are dropped.

ADA: The University of Utah seeks to provide equal access to its programs, services and activities for people with disabilities. Students are encouraged to approach the instructor and the Center for Disability Services to make suitable arrangements if needing special accommodations.