Instructor: Xuanyu Pan, pan@math.wustl.edu
TA:
Park, Jongwhan, jongwhan@wustl.edu
Office hours:
For me, Tuesday 4:005:00pm (Cupples I 107) (make an appointment)
For
TA, Monday 3:004:00pm,
Friday 3:004:00pm (Cupples I, room 207)
Required
Text: Theodore Shifrin, "Multivariable Mathematics. Linear Algebra,
Multivariable Calculus, and Manifolds", John Wiley & Sons, 2005.
Other texts:
Michael Spivak,
"Calculus on Manifolds: A Modern Approach to the Classical Theorems of
Advanced Calculus", W.A. Benjamin, 1965.
Grading: homework (15%), Midterm Exam 1 on
Monday Sep 29, 2014 (25%), Midterm Exam 2 on Friday Oct 31, 2014 (25%),
cumulative Final Exam (35%) on Dec 17 2014 10:30AM  12:30PM.
Exams will consist of a few theory questions,
including definitions and proofs of selected results, and some problems
involving computations and proofs. There will be no makeup exams  if you miss
one midterm, the final exam counts 60%. If you choose to be
graded "Pass/Fail", a "Pass" grade reqires
a grade of C or higher.
Homework: Assignments is assigned below, no
paper copy will be given in class. A grader will grade selected problems.
Discussing homework with others is ok. It is expected that everyone writes in
his/her own words the homework solutions. No late homework is accepted.
Homework is due at the beginning of class on the due date.
Prerequisites: Math 233 and Math 309 (not
concurrent), or equivalent knowledge of matrix algebra and multivariable
calculus.
Syllabus (approximate): "Vectors and
matrices. Continuity and differentiability of functions of
several variables. Partial derivatives, gradient.
Maximum value theorem. Contraction
mappings."  as in (parts of) chapters 16
of the book. Optional topics (time permitting) may include implicit function
theorem, differential forms or integration.
Tentaive outline: there will be no time to cover all of
the sections in each chapter of Shifrin's book and
not all sections will be covered in the same depth. Class time is for the
fundamental concepts of each chapter. Shifrin's book (and also the other books) are sources for further
explanations and examples. The following is the approximate plan *subject to
changes* (numbers refer to sections in Shifrin's
book); this plan will be updated as the course progresses.

(Need to hand in, check
the schedule below to figure out the deadline for each one) 
(Do not hand in, but practice by yourself or
discuss with others, they are for fun and very useful
for this course , some of them are challenging) 
Homework 1 
Section 1.1: 1(a)(d)(h), 9(a), 10 Section 1.2: 1(b), 15, 17 Section 1.3: 1(a)(b)(d), 6 
Section
1.1: 6, 7 Section
1.2: 7,8 Section
1.3: 2, 8, 9 
Homework 2 
Section 1.4: 1(a)(f), 3, 4, 8(a)(d), 23(a)(b) Section 2.1: 2(a) 
Section
1.4: 9, 10, 11, 14 Section 2.1: 1,9, 12 
Homework 3 
Section 2.2: 1(a)(e), 2, 3(a), 7(a), 12(a) Section 2.3: 8(f)(g), 12, 13 
Section
2.2: 4, 5, 6, 9(a) Section 2.3: 1, 4, 9, 11 
Homework 4 
Section 3.1: 1(a), 2(a)(d), 4 Section 3.2: 1(b)(f), 6, 18 
Section
3.1: 6, 7, 9 Section 3.2: 3(a)(c) 8,
11(a) 
Homework 5 
Section 3.3: 2, 8,
13, 17 Section 3.4: 1 (a) (b), 11 Section 3.5: 2, 3,
7(a) (b), 8(a) 
Section
3.4: 1 (b) Section 3.5: 6 
Homework 6 
Section 4.1: 14 (a), 15 (a), 17 Section 4.3: 2 (a) (e), 12(b) (c), 22(a)(b) 
Section
4.1: 3 (a) (c) (d) Section
4.3: 4, 7 
Homework 7 
Section 5.2: 1 (a),(j),(k), 13 Section 5.3: 1, 4 
Section
5.2: 2, 4 Section 5.3:
5, 6 
Homework 8 
Section 5.1: 1 (a), (b), (g), 2, 9, 10, 13 Section 6.1 1, 2, 5, 6 
Section 5.1: 5, 6, 8 
WEEK 1:
Lecture 1 (Mo Aug 25): 1.1
(vectors and matrices) Homework 1
(sec. 1.1, 1.2, 1.3)
Lecture 2 (We Aug 27): 1.1 (continuation)
Lecture 3 (Fr Aug 29): 1.2
(CauchySchwarz inequality, triangle inequality)
WEEK 2:
Monday
Sep 1st is a holiday
Lecture 4 (We Sep 3): 1.3 (subspaces of R^n)
Lecture 5 (Fr Sep 5): 1.4
(linear transformations and matrix algebra)
WEEK 3:
Lecture 6 (Mo Sep 8): 1.4
(continuation) Homework 1 due Homework 2
(sec. 1.4, 2.1)
Lecture 7 (We Sep 10): 2.1 (functions of several variables) Last day to drop (D) a FL2014 course
Lecture 8 (Fr Sep 12): , 2.2 (topology of R^n)
WEEK 4:
Lecture 9 (Mo Sep 15): 2.2 (continuation)
Last
day to change grade option on a FL2014 course to Passfail or Audit
Lecture 10 (We Sep 17 ): 2.3 (limits and continuity in several variables,
epsilondelta definition) Homework 2 due, Homework 3 (sec. 2.2, 2.3)
Lecture
11 (Fr Sep 19 ): 2.3 (continuation)
WEEK 5:
Lecture 12 (Mo Sep 22 ): 2.3 (continuation)
Lecture 13 (We Sep 24): 3.1 (direccional derivative)
Lecture 14 (Fr Sep 26): 3.2 (differentiablility)
WEEK 6:
Lecture 15 (Mo Sep 29): no
lecture, instead Midterm Exam 1 on Lectures 112 (Monday February 17, 2014) during
class time.
Lecture 16 (We Oct 1):
Correction of Midterm Exam 1 on the blackboard
Lecture 17 (Fr Oct 3): 3.2
(continuation) Homework 3 due, Homework 4
(sec. 3.1, 3.2, 3.3)
WEEK 7:
Lecture 18 (Mo Oct 6): 3.2 (continuation)
Lecture 19 (We Oct 8): catch
up
Lecture
20 (Fr Oct 10) :3.3 (Differentiation rules)
WEEK 8:
Lecture 21 (Mo Oct 13): 3.3.
(continuation)
Lecture 22 (We Oct 15): 3.4
(the gradient), 3.5 (theory of curves)
Fr
Oct 17 is Fall Break No Class
WEEK 9:
Lecture 23 (Mo Oct 20): 3.5
(continuation) Homework 4 due, Homework 5
(sec. 3.3, 3.5, 3.6)
Lecture 24 (We Oct 22): 3.6
(higher order partial derivatives, harmonic functions)
Lecture 25 (Fr Oct 24): review
of 4.1, 4.2 (linear systems of equations, inverse matrices)
WEEK 10:
Lecture 26 (Mo Oct 27):
review of 4.3, 4.4 (basis, dimension, subspaces)
Lecture 27 (We Oct 29):
review of 4.3, 4.4 (continuation), 4.5 (introduction to manifolds)
Lecture 28 (Fr Oct 31): no
lecture, instead Midterm Exam 2 on Lectures 1325 (Wednesday March 26, 2014)
during class time. Homework 5 due, Homework 6
(sec. 4.1, 4.2, 4.3, 4.4)
WEEK 11:
Lecture 29 (Mo Nov 3):
Correction of Midterm Exam 2 on the blackboard.
Lecture 30 (We Nov 5): 5.1 (compactness,convergence
theorems)
Lecture 31 (Fr Nov 7 ): 5.1 (Maximum value theorem, uniform continuity theorem)
WEEK 12:
Lecture 32 (Mo Nov 10): 5.1
(continuation)
Lecture 33 (We Nov 12): catch up and/or in class practice problems concerning
5.1 Homework 6 due, Homework 7
(sec. 5.1, 5.2)
Lecture 34 (Fr Nov 14):
catch up and/or in class practice problems concerning 5.1
WEEK 13:
Lecture 35 (Mo Nov 17 ): 5.2 (Maxima, minima, critical points)
Lecture 36 (We Nov 19): 5.2
(continuation) Homework 8
(sec. 5.1)
Lecture 37 (Fr Nov 21): 5.3
(Second derivative theorem) Homework 7 due
WEEK 14:
Lecture 38 (Mo ): 5.3 (continuation)
Nov 2630 Thanksgiving break
WEEK 15:
Lecture 39 (Mo Dec 1): 6.1
(contraction mapping theorem)
Lecture 40 (We Dec 3): 6.1
(continuation and practice problems)
Lecture 41 (Fr Dec 5): 6.1
(continuation and practice problems) Homework 8 due.