Department
of Mathematics
Mat 322: Analysis in Several
Dimensions
Spring 2006

Course instructor: David G. Ebin
(ebin@math.sunysb.edu). Phone: 28290. Office: Math Tower 5116. Office
hours: Tu,Th 1:152:15, or by
appointment.
Course Grader: Young Nam
(ynam@math.sunysb.edu). Office: Math
Tower 2117. Office hours: Tu.
2:003:00PM
and Th. 4:005:00PM or by
appointment.
Classroom: Harriman Hall 115. Time: TTh 2:203:40.
Text: Vector Calculus, Linear
Algebra, and
Differential Forms, 2^{nd} edition, by J. H. Hubbard &
B. B. Hubbard, Prentice Hall.
Description: This is a rigorous course
in Calculus of several variables. We will analyze techniques
to study functions whose domains are subsets of R^{n},
essential tools in the pursuit of advanced level mathematics, and many
sciences.
We shall begin with the geometric description of R^{n}, and
discuss matrices as linear transformations. We will develop criteria
for differentiability of functions, solve basic linear equations,
relate the dimensions of kernel and images of linear transformations,
and discuss the inverse and implicit function theorems. Then, we will
proceed with the definition
and study of higher order derivatives of functions, quadratic forms,
integration,
and submanifolds of R^{n}, the latter subject emphasizing the
case of
curves and surfaces. The course will end with a description of forms
and the
exterior differentiation operator, the statement of Stokes' Theorem and
some of its applications.
Homework: There will be homework
assignments throughout the semester, approximately weekly. No late
homework will be accepted.
Examinations: There will be a midterm
and
a final examination, as indicated below. The time of these exams is
given now so that you can resolve
any possible conflict before the course starts. There will be no makeup
exams due to conflicts.
Test 
Date 
Time 
Midterm 
March 21 
02:2003:40 
Final 
May 16 
2:004:30PM 
FINAL EXAM REVIEW: Exam
Review
This review sheet is good for the midterm as well. Go over
everything up to and including Newton's method. Stop before the
contraction lemma.
Grading: The homework assignments, the
midterm,
and the final examination will count for 20%, 30%, and 50% of your
final
grade, respectively.
SECTIONS TO BE COVERED (approximate)
Feb. 14: Sections 1.4, 1.5
Feb. 21:
Sections 1.6, 1.7
Feb. 28:
Section 1.8
March 7: Section 1.9
March 21: Sections 2.7, 2.8
March 28
April 18:
Section 2.9
April 20:
Section 3.3
April 24:
Sections 3.43.6
May 1:
Sections 3.73.8
HOMEWORK
Due: Jan 26: Read Chapter 0, except 0.6 and prepare a list
of questions on its content that you might ask in class.
Due Feb. 7:
Problems 1.1.4, 1.1.8, 1.2.2a,c,d, 1.2.4, 1.2.8, 1.2.10, 1.2.12, 1.2.13
Due Feb. 14:
Problems 1.3..4, 1.3.9, 1.3.10, 1.3.12, 1.3.14, 1.3.19, 1.3.20, 1.3.21,
1.4.3, 1.4.4, 1.4.10
Due Feb.
21: Problems 1.4.12, 1.4.16, 1.4.23, 1.4.24, 1.5.1a,b,d,f, 1.5.3,
1.5.4, 1.5.6
Due March
2: Problems1.5.8, 1.5.10, 1.5.13, 1.5.20, 1.5.21a,b, 1.6.3, 1.6.5,
1.6.10, 1.6.11
Due March 14:
Problems 1.7.2, 1.7.6, 1.7.10a, 1.7.12, 1.7.19, 1.7.22
Due March 30:
Problems 1.8.2, 1.8.6, 1.8.10, 1.9.2, 2.7.1,
2.7.3, 2.7.6, 2.7.9, 2.7.13 a,b
Due April 7:
Problems 2.8.2, 2.8.3, 2.8.4, 2.8.5, 2.8.8, the last problem is
difficult
Due April 18:
Problems 2.9.2, 2.9.3, 2.9.5, 2.9.6, 2.9.8, 2.9.11, 2.9.12, 2.9.15,
2.9.7
Due April
27: http://www.math.sunysb.edu/~daryl/322prob.pdf
Due May 4:
http://www.math.sunysb.edu/~daryl/322prob2.pdf
If you have a physical, psychological, medical or learning
disability that
my impact on your ability to carry out assigned course work, please
contact
the staff in the Disabled Student Service Office, Room 133, Humanities,
6326784/TDD. DSS will review your concerns and determine with you what
accommodations are necessary and appropriate. All information and
documentation of disability is confidential.

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