Fall 2013

**Section 1: MW 8:40am - 9:55am**, Mathematics 312

**Section 2: MW 10:10am - 11:25am**, Mathematics 312

*All important course announcements will appear in Courseworks.*

**Instructor:**
Michael Woodbury

Phone: x4-4988

Office: 427 Mathematics

email: MYLASTNAME at MATH dot COLUMBIA dot EDU

**Office Hours:** Friday 4:30pm-6:00pm, Monday 1pm-2:30pm or by appointment.

**Teaching Assistant:**

Undergrad TAs: Min-hwan Oh, Siqi Cao

Grad TAs: Tristan Collins, Pak Hin Lee

**Text:** James Stewart * Calculus: Early Transcendentals*, seventh edition, Brooks/Cole, 2008.

**Course description:** We will cover chapters 15 (Multiple Integrals) and 16 (Vector Calculus). The main topics are:

- multiple integrals (using rectangular coordinates)
- integrals using polar, cylindrical, and spherical coordinates
- vector fields
- line integrals
- gradients, curl, and divergence.
- Green's Theorem, Stoke's Theorem, Divergence Theorem

We will also cover topics from basic complex analysis. The required material will be presented in class. Additional reference materials will also be provided.

Here are course notes written by Prof. Herve Jacquet. This (together with the lectures) is the main reference for the section on complex numbers.

Complex Numbers

Complex Functions and the Cauchy Riemann Equations

Contour Integrals and Cauchy's Theorem

If you feel like you would benefit from additional reading material, here is an online book on complex analysis:

Complex Analysis by George Cain

It is pretty readable and has a number of good exercises for practice.

**Prerequisites: **All material covered in Calculus I-III (Stewart chapters 1-14, except those involving differential equations) will be assumed. In particular, knowing the theory of integration in one variable well will be especially important.

**Important Dates:**

- September 3: First day of classes
- October 8: Drop date (most colleges)
- October 2: Midterm 1 (This is the
*tentative*date and is subject to change.) - November 4-5: University holiday (Remember to vote)
- November 13: Midterm 2 (This is the
*tentative*date and is subject to change.) - December 9: Last day of classes
- December 16 or 18: Final exam. (The final exam for section 2 is on Monday, and section 1 is on Wednesday.)

**Advice:** Previous experience suggests that students find Calculus IV much more challenging than Calculus I-III. Do everything you can to not fall behind. Here are some suggestions:

- Be an active learner. Watching me or a TA do math without doing work on your own will not be enough to be able to reproduce the work yourself on quizzes or exams.
- Ask questions in class, and utilize office hours if needed.
- Read the book. Look at the book
*before*class to get an idea of the material we are going to cover, then reread it more carefully*after*class. - Attend class. (Not everything we cover will be from the book).
- Do the homework. Besides being graded, this is your best resource for learning the math. Also, keep in mind that the point of homework is
*not*to get the right answer; rather, it is a learning tool. As such, don't cheat yourself by racing through the problems or leaving out justification of each step. Also, be sure to avoid relying on a solutions manual-resources such as a solution manual can give the false impression of understanding. - Do problems that aren't assigned.

**Resources:** Help is always available from the Help Room (Mathematics 406,
10am-5pm Mon-Fri), me, or the TAs.

If there's something you don't understand, ASK.

**Written Homework:** Written homework will be assigned every week except when there is a Midterm. This means there will be 12 homework assignments. Assigned problems can be found in the schedule below. Problems in parentheses are optional.

- Write your name, your uni, Calculus IV Section #(1 or 2 as appropriate), and my last name (Woodbury) in the top right hand corner of your assignment.
- It will be collected on Mondays. Please give homework to me directly or deposit it to the dropbox (located on the fourth floor of the mathematics building.) DO NOT put homework in my mailbox since it may not reach me in time this way.
*Late homework will not be accepted.*- Collaboration is encouraged, as long as the work you hand in is your own. Also, staple your homework.
- For all written work, solve the problems in an organized fashion, with clear explanations. The grader can't/won't read your mind.
- 20% your homework grade will be based on neatness and completeness. Neatness means: do not turn in pages with frayed edges (i.e. from a spiral bound notebook); staple neatly; your work must be legible. Completeness means: try all of the problems.
- Knowing how to solve the problems like those which are assigned will be to your benefit when taking exams.

**Web-based Homework:** You are also responsible for WebAssign homework. This is an online homework system that will give you the instant feedback of knowing whether your answers is correct or not.

- To register for WebAssign: 1) Go to the site. 2) In the "Account Log In" area, click on "I have a class key." 3) Enter the class key: columbia 8722 8737. 4) If you don't have one already, create an account. (Use your uni as your username.)
- If you purchased your textbook from the Columbia bookstore and still have the access code, you'll be able to enter the code and be set to do homework. You'll also be able to view an electronic copy of the textbook. If you don't have an access code, you can sign up for a limited time free trial period.
- The first WebAssign homework will be posted on September 4. It will be due September 9 at 11:59pm. All future problem sets will be due by 11:59pm Monday evenings.
- Questions related to WebAssign should be directed to the graduate TA. (For assignments 1-6, send questions to Pak-Hin. For assignments 7-12, contact Tristan.)
- As with written homework the 10 best scores will count towards your grade.

**Exams:** We will have 2 midterms, and one final

- See schedule below for dates. Note that these dates are tentative and may change.
- The final exam will be on December 16 (for section 2) and December 18 (for section 1.) All exams will be given in class.
- Calculators are NOT allowed on exams.
*There will be no make-up exams without a note from a doctor or a dean.**Let me know immediately if there's going to be a conflict*

**Grading:** The grading scheme is as follows:

- Final: 30%
- Midterms: 25% (each)
- Homework: 20%

Date | Topics/Sections covered | Remarks | Homework | Date Due |
---|---|---|---|---|

September 3 | Cartesian, polar, cylindrical and spherical coordinate systems | HW1: 10.3 #7-12,(54); 15.8 #8,12,13; 15.9#14,15 | September 9 | |

September 9,11 | (15.1-3) Integrals over rectangles and more general regions | HW2: 15.1 #2,(8,10),14; 15.2 #24,(40) 15.3#12,14,16,38,(56,62) | September 16 | |

Sept 16,18 | (15.4-5) Polar Coordinates; Applications; Triple Integrals | HW3: 15.4 #1-4,6,10,14,(39,40); 15.5 #28,(32); 15.6#(22,24) | September 23 | |

Sept 23,25 | (15.6-8) Cylindrical and spherical coordinates | HW4: 15.7#28,30,34,48,(55a); 15.8#16,30 | September 30 | |

Sep 30, Oct 2 | (15.9) Change of coordinates; Review | |||

Oct 7,9 | (16.1) Midterm 1; Vector Fields | HW5: 15.9#18,20,(47); 15.10#8,14; 16.1#4,6,26,(35),36 | October 14 | |

Oct 14,16 | (16.2-3) Line Integrals; Fundamental theorem for line integrals | HW6: 16.2#18,34,40,(45,49,50); 16.3#11,26,28,(29,35) | October 21 | |

Oct 21,23 | (16.4-6) Green's theorem; Curl and divergence | HW7: 16.4#2,19,(21,22),28; 16.5#9-11,12,20,(30,32,37,39); 16.6#20,(22),24,(64) | October 28 | |

Oct 28,30 | (16.7-8) Surface integrals; Stoke's theorem | HW8: 16.7#39,40,44; 16.8#16,17,(20) | November 6 | |

Nov 6 | (16.9) Divergence theorem | HW9: 16.9#2,4,18,(19),20,24 | November 11 | |

Nov 11,13 | Review; Midterm 2 | |||

Nov 18,20 | Complex Functions 1 | HW10: The problems from Complex Numbers notes. | November 25 | |

Nov 25 | Complex Functions 2: Cauchy-Riemann equations | HW11: Problems from Complex Functions and the Cauchy Riemann Equations plus extra problems. | December 2 | |

Dec 2, 4 | Complex Functions 3:Contour integrals and Cauchy's theorem | HW12: Problems from Contour Integrals and Cauchy's Theorem | December 9 | |

Dec 10 | Review | |||

December 16/18 | Final Exam |

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