MATH 318 : Calculus of Several Variables (Fall 2014)


Instructor: Xuanyu Pan, pan@math.wustl.edu

TA: Park, Jongwhan, jongwhan@wustl.edu

Office hours:

For me, Tuesday 4:00-5:00pm (Cupples I 107) (make an appointment)

For TA, Monday 3:00-4:00pm, Friday 3:00-4:00pm (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 make-up 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 1-6 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 (Cauchy-Schwarz 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 Pass-fail or Audit

Lecture 10 (We Sep 17 ): 2.3 (limits and continuity in several variables, epsilon-delta 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 1-12 (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 13-25 (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 26-30 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.