Math 318 Fall, 2013

Review session for the Final: Dec 16 Monday 5:30-7:30 at UMRATH 140

Office hours for the Final: Dec 9 Monday 4:00 - 5:30, Dec 16 Monday 2:00 - 4:00, or by appointment

Final Exam: Dec 18 Wednesday 1:00-3:00 at Wilson 214

Here is a.

Lectures: MWF 1:10-2 Wilson 214

Instructor: Songhao Li        Office: 207A Cupples I

Phone: (314)935-4208        email: sli@math.wustl.edu

Office Hour: M 2:30-4, F 11-12

Course Webpage: http://www.math.wustl.edu/~sli/math318_fall2013/course.html

Textbook: Multivariable Mathematics by Theodore Shifrin

Suggested: Calculus on Manifolds by Michael Spivak (This text could be useful if you have taken Math 310.)

After a brief review of linear algebra of Euclidean space, we study calculus of several variables with some rigour. Topics may include:

-- Limits and continuity;

-- Derivatives and extrema;

-- Integrations;

-- Introduction to manifolds.

Homework            20%

Test 1 (Oct 7)        20%

Test 2 (Nov 11)     20%

Final (Dec 18)        40%

Tests will be in class and of 45 minutes in length.

Homeworks:

While collaboration is permitted, and in fact, encouraged, one should write up his/her own solution.

In other words, the exchange of ideas is permitted, but sharing the submitted version is not.

Please note that the grader and I will take notice of striking similarity in writing and take approriate action.

Assignments are due at the beginning of the class.

It is okay if you are late by a few minutes, but no late assignments will be accepted unless you have a valid reason.

If you cannot make it to a class when an assignment is due, you may arrange for a friend to submit it for you.

There will be about 8 assignments, to be posted here. Hand-written solutions are provided for your convenience. They have not been proof-read and may contain minor errors.

Week 1

Aug 28:  vectors in R^n Homework 1

Aug 30:  dot product and subspaces of R^n

Week 2

Sept 2:  Labour Day - No Class

Sept 4:  linear transformation and matrix multiplication

Sept 6:  vector-valued function of several variables and topology of R^n

Week 3

Sept 9:  topology of R^n continued Homework 2

Sept 11:  topology of R^n continued

Sept 13:  function limit

Week 4

Sept 16:  function continuity

Sept 18:  function continuity continued Solution 1

Sept 20:  directional direvative

Week 5

Sept 23:  direvative Solution 2

Sept 25:  direvative continued Homework 3

Sept 27:  direvative continued

Week 6

Sept 30:  direvative coninued

Oct 2: differentiation rules

Oct 4: review Solution 3

Week 7

Oct 7:  Test 1 Test 1 with solution

Oct 9: differentiation rules continued (chain rule) Homework 4

Oct 11: differentiation rules continued (appliations of chain rule)

Week 8

Oct 14: Test 1 reviewed

Oct 18: Fall Break - No Class

Week 9

Oct 21: gradient continued Solution 4 Homework 5

Oct 23: curves - arclength

Oct 25: curves - curvature

Week 10

Oct 28: higher order partial derivatives

Oct 30: higher order partial derivatives Solution 5 Homework 6

Nov 1: Compactness and Maximum Value Theorem

Note: We will not cover Chapter 4 in class. Instead, we will pick up linear algebra results as needed.

Week 11

Nov 4: Maximum/Minimum Pratice Test 2

Nov 6: Second derivative test - examples

Nov 8: Review Solution 6

Week 12

Nov 11: Test 2 Test 2 with solution

Nov 13: Second derivative test - proof Homework 7 (updated, removed 5.2.3)

Nov 15: Second derivative test - examples and counter examples Term grades

Note: Differing from Shifrin, we use the signs of the eigenvalues of a symmetric matrix to determine if the corresponding quadratic form is positive definite, negative definite or indefinite.

Week 13

Nov 18: Lagrangian multipliers - proof

Nov 20: Lagrangian multipliers - examples

Nov 22: Contraction mapping principle - examples Solution 7

Week 14

Nov 25: Contraction mapping principle - proof Homework 8

Nov 27: Happy Thanksgiving - No Class

Nov 29: Happy Thanksgiving - No Class

Week 15

Dec 2: Inverse function theorem

Dec 4: Inverse function theorem - proof

Dec 6: Implicit function theorem Homework 9 (Not to be handed-in) Solution 8 Solution 9