THE
UNIVERSITY OF HAWAI'I AT HILO - MATHEMATICS DEPARTMENT
Course
Description:
Real Analysis I and II focus on an analytic study of the real line R
and its higher dimensional analogue R^n. To many students,
Analysis is simply a more detailed look at the various concepts and
theorems from Calculus. While this may be a major motivation, it
is not until one looks deeply into the Topology of the Real line that
one sees that this seemingly ordinary object is remarkably deep and
complicated. Even answering the question of "What is a real
number?" will be difficult to answer at first. The purpose of
this course is to take an axiomatic approach at studying the real line
and continuous/differentiable functions on it.
Learning
Outcomes:
The successful student will be able to:
- write a rigorous analytic proof, complete with full syntax and proper
mathematical notation
- prove that a number is or not rational
- prove various properties about integer, rational, irrational, and
complex numbers
- understand the basic concepts of the Complex plane
- use basic properties of sets and functions on sets
- utilize the basic concepts in point-set topology (open/closed sets,
compactness)
- prove that a sequence converges or diverges
- prove that a function is or is not continuous
- understand the basic properties of derivatives
Homework:
This
course will have frequent homework assignments. In fact, homework
will be the primary pedagogical tool in this course. Since
mathematics is a highly communal activity, you are encouraged to work
in groups. That being said, copying
will be severely punished. If you write something down on
your HW and can't explain it to me later on in person, then I'll
consider this to be copying.
All homework is required to be typeset in LaTeX. Homework must be
printed out prior to class and turned in at the
beginning of lecture on the due date. LaTeX:
The
use of AMS LaTeX is not only encouraged but will be
required. LaTeX is the
universally accepted scientific typesetting program that allows
mathematicians to write well-formatted mathematical papers. This
open-source compiler and associated interfaces can be found as
freeware. Please see the below links for how to download the
appropriate software.
All HW assignments can be found on our HW site in .pdf form. If
you change the ".pdf" to ".tex", your browser will download the LaTeX
source code.
The final
grade will be based largely on homework, midterms, and final.
Furthermore, class participation (e.g., asking questions
in
class, going to the Math Lab, going to
office hours, asking Bob email
questions, consistent attendeance in class, etc) will also be used in
this computation. The class participation grade is computed at
the discretion of the instructor.
Your
grade will be computed using the following weights:
50% - Homework
40% - Midterms & Final 10% -
Participation