Fall 2011 MATH 115
Functions of a Real Variable
MWF 10:10-50 am. Room: Herrin 185.
Instructor:
Isabelle
Camilier
Room
383 FF (Math Building)
Email:
camilier (at) stanford (dot) edu
Office
hours:
Monday
11-12:30 and Tuesday 1-2:30 pm,
or by appointment. Please put 'MATH 115 your request' in the object of your email.
Course
Assistant:
Alexandr Zamorzaev
Email: alzaor
(at) math (dot) stanford
(dot) edu
Office hours: Monday 1:30-3:30 pm and
Tuesday 10-11 am.
Room: 380M.
Course Description
The
development of real analysis in Euclidean space: sequences and
series, limits, continuous functions, derivatives, integrals. Basic point
set topology.
Textbook: Elementary Analysis:
The Theory of Calculus by Kenneth A. Ross (Undergraduate Texts in Mathematics,
Springer Verlag, 1980).
We
cover sections
1,2,3,4,5,7,8,9,10,11,12,14,15,17,18,19,20,23,24,25,26,28,29,30,31,32,33,34
of the book.
Assignments
Homework (20%). There are 8 problem
sets. No late homework but I drop the 2
lowest grades (only the 6 best grades count).
Midterm exam (30%) in class. Friday October 28. 10 a.m -noon ( 2 hours). Room 420-041 (not the usual room). Alternate time 9 a.m.-11 a.m.
Final
exam (50%) in class.
Thursday December 15, 8:30-11:30 am. Room 380-380C (basement
of math building).
Syllabus
Week
1 and 2: Introduction. Basic properties of sequences.
Week3:
Monotone sequences. Cauchy sequences.
Week4:
Series.
Week5:
Series. Continuity.
Week6:
Continuity. Uniform continuity. Limits of functions.
Week7:
Uniform convergence of sequences of functions. Series of
functions.
Week8:
Integrals. Power series.
Week9:
Differentiation. Taylor's Theorem.
Week10:
Review.
Homework
HW1
Average:
18/20. Stdev 2.4.
Schedule
|
|
Topics |
Textbook |
Notes |
|
Monday 09/26 |
Introduction.
Mathematical induction. |
1 |
|
|
Wednesday 09/28 |
Induction. Rational
numbers. |
2 |
|
|
Friday 09/30 |
Real numbers. Field
properties. Triangle inequality |
3 |
|
|
Monday 10/03 |
Real numbers, sup, inf |
4,5 |
|
|
Wednesday 10/05 |
Real
numbers. HW1 due |
4,5 |
|
|
Friday 10/07 |
Sequences:
definitions. |
7,8 |
|
|
Monday 10/10 |
Sequences |
8,9 |
|
|
Wednesday 10/12 |
Sequences. HW2 due |
9 |
|
|
Friday 10/14 |
Sequences |
10 |
|
|
Monday 10/17 |
Monotone
sequences. |
10 |
|
|
Wednesday 10/19 |
Sequences.
limsup, liminf HW3 due |
10,12 |
|
|
Friday 10/21 |
Sequences.
Subsequences. Bolzano-Weierstrass. |
11 |
|
|
Monday 10/24 |
Series |
14,15 |
|
|
Wednesday 10/26 |
Midterm
review. |
17,20 |
|
|
Friday 10/28 |
Midterm exam. |
|
|
|
Monday 10/31 |
Series |
14,15 |
|
|
Wednesday 11/02 |
Series.
Open intervals. HW4 due |
15,
21 |
|
|
Friday 11/04 |
Limits.
Continuity: definition. |
20,17 |
|
|
Monday 11/07 |
Continuity |
17 |
|
|
Wednesday 11/09 |
Continuity. HW5 due |
17,18 |
|
|
Friday 11/11 |
Uniform
continuity. |
19 |
|
|
Monday 11/14 |
Uniform
continuity. Sequences and series of functions. |
19,
24 |
|
|
Wednesday 11/16 |
Convergence
of power series. HW6 due |
23 |
|
|
Friday 11/18 |
Uniform
convergence, more on power series |
24,25 |
|
|
Thanksgiving break |
|||
|
Monday 11/28 |
Power
series. Integration of power series |
26, |
|
|
Wednesday 11/30 |
Integration. HW7 due |
26,
32-34 |
|
|
Friday 12/02 |
Differentiation |
28,29,30 |
|
|
Monday 12/05 |
Differentiation
(of power series). TaylorÕs
theorem. |
26 |
|
|
Wednesday 12/07 |
Review.
HW8 due |
31 |
|
|
Friday 12/09 |
Review. |
|
|
|
Thursday 12/15 |
Final exam. 8:30-11:30 am |
||