MA 281, Spring 2006
HONORS MATHEMATICAL ANALYSIS III



General information

Description: This is the second installment of a 3-course sequence in mathematical analysis for beginners. For a description of the style and format of this course see the General Information section of the MA 182, Fall 05, Web Page. The course is intended for math majors and students with a serious mathematical interest. Successful completion of MA 182 or permission by the instructor are required for admission.

Lectures: Monday, Wednesday 12:00-12:50, E. A. Stevens 329.

Recitations: Friday 3:00-3:50, E. A. Stevens 229.

Lecturer: Marco Lenci. Office: Kidde 219B. Tel.: x-5453. E-mail: mlenci@math.stevens.edu.

Teaching Assistant: Luca Bussolari. Office: Kidde 105. Tel.: x-5431. E-mail: lbussola@stevens.edu.

Office hours: Prof. Lenci's official office hours are posted here (but students are welcome to visit at other times as well, by appointment). Mr. Bussolari's office hours are TBA.

Textbooks: T. M. Apostol, Calculus, Vol. I & II, 2nd ed., Wiley.


Grades and policy

Attendance: Attendance is very strongly recommended but not enforced. It is every teacher's experience that students who attend the lectures tend to do significantly better in a course than student who don't. However, students in this course will be considered responsible for managing their own time. (Besides, enrolling in an honors class and not attending seems contradictory.) As for missing quizzes or exams, see the policy on make-ups. (See also the last remark in the Final grade section.)

Homework: The syllabus contains a list of suggested homework problems from the textbook, for every week. The student is strongly adviced to work out as many as they can from the list. However, no homework is due or counts for the final grade. Nonetheless, students who want their homework graded and commented on can turn it in or, even better, discuss it with Prof. Lenci or Mr. Bussolari during their office hours.

Quizzes: There will be a 10-minute quiz (almost) every week, on topics up to the previous week. The quiz will take place at the beginning of class on Wednesday. Students can miss up to 2 quizzes with no justification required (the worst 2 quiz grades will be discarded in the final average). Beyond that, see the policy on make-ups.

Exams: There will be a certain number of 55-minutes midterm exams and a 4-hour final exam. Dates will be announced during the semester. The midterm exams will take place on Wednesday during class time.

Make-ups: Make-up quizzes and exams are allowed only in exceptional cases, which must be motivated (possibly with the proper documentation) and communicated to Prof. Lenci as soon as possible. Failure to address any of these points will result in the request being denied and a grade of 0 being given for the corresponding test.

Final grade: The final average will be computed as follows

   25%   Quizzes (worst two grades discarded)
   40%   Midterms
   35%   Final

and then converted to a letter grade using the following scale

Grade F D C- C C+ B- B B+ A- A
% Avg 0-50 50-60 60-64 64-70 70-74 74-78 78-82 82-86 86-90 90-100

In exceptional cases, the resulting final grade may be raised based on special considerations, such as a marked improvement of the student during the semester, a significant health problem or similar distressing situation affecting an exam, etc. In these cases, the diligence displayed by the student during the semester (including attendance and class participation) makes it easier for Prof. Lenci to be generous.


Syllabus

Note: The topic and homework lists are not considered definitive until after Wednesday of the current week. (Data in italics is merely the plan, and are subject to change.)

Week Topics Book Sections Homework Notes
1/18-20 Review of the logic of quantifiers.
Integration by parts.
5.9 5.10, 1-9  
1/23-27 Uniform continuity.
Integrability of continuous functions.
3.17-18 HW Sheet #1  
1/30 - 2/3 Taylor polynomials and error estimates.
Landau symbols (o and O).
Applications to indeterminate forms.
7.1-2
7.5-6
7.9-10
7.4 #1,2,4,6,9,10
7.8 #1,2,8
7.11 #1,2,5,6-10,12-17,30,32
 
2/6-10 Hyperbolic functions and inverse trig functions
Improper integrals.
6.18,21
10.23
6.19 #1-6
6.22 #1-3,6,9,12-18,29-35
 
2/13-17 Comparison test for improper integrals.
Series. Geometric and telescoping series.
10.23
10.5-6
10.7-8
10.24 #1-8,11,16,19,20,21
10.9 #1-5,9
Bussolari takes over
2/21-24 Convergence. The comparison test.
The integral test.
The ratio test.
10.11-13
10.15
10.14 #1-5,11-14,16
10.16 #1-6,9-11,13
Tue Feb 21 Monday schedule
(President's Day)
2/27 - 3/3 Review.
The alternating test.
Absolute Convergence.
10.17-18 10.20 #1,2,3-9,15,16,20,29,30 Lenci back
Midterm #1 on Mar 1
3/6-10 Sequences of functions.
Pointwise and uniform convergence.
Weierstrass Test for series.
Power Series.
11.1-2
11.5
11.6, 11.8
HW Sheet #2
11.7 #1-8,17-18
10.9 #11-17
Spring Break: Mar 12-19
3/20-24 Differentiation and integration of power series.
Taylor series and their convergence.
11.9-11 HW Sheet #3
11.13 #11-13,15
 
3/27-31 An example of a Taylor series not converging to its function.
Ordered pairs. Vectors as n-tuples or reals. Basic operations.
Geometric vectors.
Vector-valued functions.
12.2
12.3
14.1-3
12.4 #5,6,9,10
14.4 #1-4,8-10,14,19
 
4/3-7 Scalar (or dot) product. Norm (or length) of a vector.
Orthogonality. Angle between two vectors.
12.5-6
12.7, 12.9-10
12.8 #2,3,4,5,10,19,20
12.11 #3,7,8
Midterm #2 on Apr 5
Lecture on Friday Apr 7
4/10-12 Velocity, acceleration.
Unit tangent, principal normal.
14.5-6
14.8
HW Sheet #4
14.7 #1-4,7,10
14.9 #1-4(a only), 7(a,c,d)
No recitation on Apr 14
(Good Friday)
4/17-21 Scalar and vector fields. Graphical representations.
Interior points. Neighborhoods. Open and closed sets.
Limits for many variables. Properties and conterexamples. Continuity.
8.1-2
8.4
8.3 #1(a,c,d), 2(a,c,d,g,j,l), 3(a-d), 5(b-e), 6(a-f), 8(a,c-e)
8.5 #1(a-d,h,j), 3, 4, 6, 7
Henceforth references are to Apostol, Vol. II
4/24-28 Review.   Review Sheet for Midterm #3 Midterm #3 on Apr 28
5/1-3 Derivatives w.r.t. a vector and directional derivatives. Partial derivatives.
Directional derivatives and continuity.
Elements of linear transformations.
The total derivative (differential). Gradient of a scalar field.
8.6-8
8.10-12
8.9 #1,4-6,8,10-12,18
8.14 #1,4,7(a,d)
Wed May 3 Friday schedule

Final Exam: May 9, 8:00-12:00, in P-216.


Communications to the class

May 11: Final class grades are posted. Prof. Lenci available during the week of May 15 to view the final.

May 2: Review session on Friday May 5, 11:00am-1:00pm, in E-229A.
Apr 19: Midterm #3: Friday Apr 28, in class.
Mar 6: Midterm #2: Wednesday Apr 5, in class.
Feb 16: Midterm #1: Wednesday March 1, in class.
Feb 16: Registrar communicated Final Exam Date: May 9, 8:00-12:00, in P-216.
Jan 25: Class couldn't agree on recitation schedule change. Will remain F 3:00-3:50 in E-229.
Jan 19: Asked the registrar to move Friday class to earlier day/time.


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