Math 318 Spring, 2015

Lectures: MWF 2:10pm-3:00pm Room 230 Cupples II

Instructor: Songhao Li        Office: 207A Cupples I

Phone: (314)935-4208        email:

Office Hour: MW 1pm-2pm or by appointment

TA: Jongwhan Park

TA office hour: MW 4:30pm-5:30pm Room 8 (basement) Cupples I

Course Webpage:

Textbook: Multivariable Mathematics by Theodore Shifrin

Suggested: Calculus on Manifolds by Michael Spivak

Note: Spivak is a more advanced and more condensed textbook. It is useful if you plan to pursue graduate studies in mathematics or math-related subjects.

We will cover most, if not all, of the following topics:

-- Linear algebra of Euclidean space;

-- Limits and continuity;

-- Derivatives and extrema;

-- Inverse and implicit function theorems;

It is unlikely that we will have time to cover integration, but for your own sake, it will give a more complete picture if you read the chapters on integration.

For most of you, even though the topics sound familiar, the course will feel significantly different from the previous calculus courses you have taken.

Everything will be defined and proved with rigour.



Homework            20%

Test 1 (Feb 11)        20%

Test 2 (Mar 18)     20%

Final (May 4)        40%

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



While collaboration is permitted, one should write up his/her own solution.

That is, the communication 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 approximately 9 assignments, to be posted here.

The progress of the course will also be documented.

Progress of the course:

Week 1

Jan 12:  vectors in R^n, dot product, subspaces of R^n Assignment 1

Note: Assignment 1 is a bit longer, but you also have two weeks to complete it.

Jan 14: linear transformation, matrix

Jan 16:  functions on  R^n

Week 2

Jan 19: No class

Jan 21: topology of  R^n: open sets

Jan 23: topology of  R^n: closed sets Assignment 2

Week 3

Jan 26: topology of  R^n: the complement of a closed set is open

Jan 28: limits

Jan 30: continuity Assignment 3

Week 4

Feb 2: partial derivative

Feb 4: derivative

Feb 6: examples of derivative Assignment 4

Note: Assignment 4 is due on Feb 20!

Week 5

Feb 9: review for test 1

Feb 11: test 1

Feb 13: continuous partial derivative implies differentiablity Test 1 with solution

Week 6

Feb 16: differentiation rules

Feb 18: solution for test 1

Feb 20: chain rule Assignment 5

Week 7

Feb 23: gradient

Feb 25: Kepler's 2nd law

Feb 27: arclength Assignment 6

Week 8

Mar 2: curvature

Mar 4: Kepler's 1st law

Mar 6: Kepler's 3rd law (cf. Exercise 3.5.15)

Week 9

Spring Break

Week 10

Mar 16: Review for Test 2

Mar 18: Test 2 Test 2 with solution

Mar 20: Higher order partial direvatives Assignment 7

Week 11

Mar 23: Harmonic functions, Laplace's equation and wave equation

Mar 25: Compact subset of R^n

Mar 27: Maximum value theoreom and the norm of a linear map Assignment 8

Week 12

Mar 30: Uniform continuity

Apr 1: Critical points and examples

Apr 3: Hessian matrix and Tayler polynomial up to order 2 Assignment 9

Week 13

Apr 6: Second derivative test

Apr 8:  Examples of 2nd derivative test

Apr 10: Lagrangian multiplier Assignment 10

Week 14

Apr 13: Proof of Lagrangian multiplier

Apr 15: Examples of Lagrangian multiplier

Apr 17:  Contraction mapping principle Assignment 11 (updated: removed 6.1.8)

Week 15

Apr 20: The mean value inequality, Newton's method

Apr 22: Inverse function theorem

Apr 24: Proof of inverse function theorem, implicit function theorem Assignment 12 (not to be handed in)

Final Exam wit solutions (Updated, corrected typos and mistakes!)