Rigorous
presentation, with proofs, of fundamental concepts of analysis:
limits, continuity, differentiation, integration, fundamental theorem,
power series, uniform convergence.
The origins of some ideas of mathematical analysis are lost in
antiquity. About 300 years ago, Newton and Leibniz independently
created
the calculus. This was used with great success, but for the most part
uncritically, for about 200 years. In the last century, mathematicians
began to examine the foundations of analysis, giving the concepts of
function, continuity, convergence, derivative and integral the firm
basis
they required. These developments continue today, for example with the
study of calculus on infinite dimensional spaces.
This course is a continuation of first and second year calculus. The
material will be presented in a rigorous manner, putting emphasis on
careful mathematical arguments, proofs and illustrative examples. The
objectives of the course include improved skill at understanding and
writing mathematical arguments as well as understanding of the
concepts of the course.
The course will provide a useful theoretical background for a
variety of higher level and graduate courses including those in
analysis,
probability, topology, mathematical statistics, and numerical
analysis.
The textbook will be Bartle and Sherbert, Introduction to Real
Analysis.
Prerequisite:At least one of the following four courses or
course combinations: (1) AS/SC/MATH 2010 3.0, (2) AS/SC/AK/MATH
3110 3.0, (3) AS/SC/AK/MATH 2310 3.0 and AS/SC/MATH 1010 3.0, (4)
AS/SC/MATH 2015 3.0 and AS/SC/MATH 1010 3.0.
Coordinator: T. Salisbury
