Spring 2015

**Section 3: TR 11:40am - 12:55pm**, Mathematics 312

**Section 4: TR 1:10pm - 2:25pm**, 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 9am-10:30am; Tuesday 2:30pm-4pm; or by appointment.

**Teaching Assistants:**

Undergrad TAs: Greyson Potter, Riaz Helfer

Grad TAs: Natasha Potashnik, Zhijie Huang

**Text:** James Stewart * Calculus: Early Transcendentals*, seventh edition, Brooks/Cole, 2008. More information about the textbook can be found on the math department's Calculus page

**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. Additionally, here are some 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 will be especially important.

**Important Dates:**

- January 20: First day of classes
- February 19: Midterm 1
- February 24: Drop date (most colleges)
- March 16-20: Spring Break
- March 26: Last Day to Pass/Fail
- April 2: Midterm 2 (This is the
*tentative*date and is subject to change.) - May 4: Last day of classes
- TBD: Final Exams (The projected schedule has section 4 at 1:10pm on Tuesday, May 12 and section 3 at 410 on Thursday, May 14.)

**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, a TA or another student do math without doing work on your own will not be enough to be able to reproduce the work on exams.
- Ask questions in class, and utilize office hours if needed.
- Read the book. Besides having many worked out examples, the book provides explanations of how the formulas are derived. Understanding this material makes doing the problems much easier. Also, the ebook (available through webassign) has additional resources not available in the physical text. For example, there are links video presentations.
- Attend class. (Not only can you receive credit towards your final grade by participating in class everyday, but it will provide you the opportunity to discuss the material with your peers and better understand the concepts involved.
- Do the homework. Besides being graded, this is your best resource for learning the math. 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 resources--such as a solution manual or a TA/tutor--which you won't have at your disposal in the exams. These resources, although valuable if used correctly, can give the false impression of understanding if abused. - Do problems that aren't assigned.

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

If there's something you don't understand, ASK. Besides going to the help room and/or coming to my office hours, you can ask questions (anonymously, if you wish) through Piazza.

Some information about free/cheap tutoring.

**Daily Homework:** Written homework will be assigned every day except, possibly, when there is a Midterm or a review session. Assigned problems can be found in the schedule below.

- Write your name, your uni, Calculus IV Section #(3 or 4, as appropriate), and my last name (Woodbury) in the top right hand corner of your assignment. Center a title at the top of the assignment: Daily Homework #(as appropriate). You are welcome to write down the specific problems as part of the title as well.
- Use pencil or black ink for your homework
- Homework will be collected at the end of class, but it will only be graded on what you have completed
*beforehand*. You are on your honor to not alter your solutions during class. You may add notes to your work during class if you use a (not red or black) pen. For example, blue, purple or green pens are acceptable colors for your annotations. These notes will not effect the score of your homework (for better or worse.) - If I suspect cheating, the priveledge of retaining your homework throughout class may be revoked. If you choose not to (or cannot) come to class, you may deposit homework in the dropbox (located on the fourth floor of the mathematics building)
*before*class. No late homework will be accepted. - The lowest three scores will be dropped from the final grade.
*Late homework will not be accepted.*- Collaboration is encouraged, as long as the work you hand in is your own.
- For all written work, solve the problems in an organized fashion, with clear explanations. It should be written in a way that the grader need not look at the book or read between the lines in order to follow what is being done.
- A portion of your homework grade will be based on neatness. Neatness means: do not turn in pages with frayed edges (i.e. from a spiral bound notebook); staple neatly; your work must be legible; it must not be overcrowded. Write your solution using pencil or black pen.
- A portion of your homework grade will be based on completeness. This means that you will receive credit for adequately attempting to solve each problem.
- The rest of the homework grade will be based on carefully grading a subset of the problems. Your presentation, as well as the correctness, of the solution will be assessed. Part of the presentation includes adequately writing down the statement of the question.

**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 upper right hand corner of the page, click on "Enter class key." 3) Enter the class key: columbia 0117 0941. 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. In any event, you can use WebAssign for free for two weeks after which time continued access for a single semester costs $75.
- In WebAssign, 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, but eventually, you will need to pay for the right to use WebAssign.
- There will be a lag between when daily homework is due and when WebAssign is due. This should give you more than enough time to complete the assignments. As such, no extensions will be given.
- The lowest score will be dropped in calculating your final grade.

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

- See schedule below for dates. I don't expect these dates to change, but for the time being they are only tentative.
- The time and place of the final exam is yet to be determined for certain.
- 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 or if you need special accommodations in any way on the exams.*

**Participation:** You will get credit for coming to class and being actively involved.

- In class polls/quizzes: By responding to class polls and quizzes (whether your answers are correct or not), you will receive credit for having come to class. Missing one or two days throughout the semester shouldn't negatively impact this portion of your participation score.
- Board Work (and follow-up): I will select students each day to write up their solutions to the daily homework. These students will then be responsible for producing typed solutions to the same problems.
- Group Work: During class, you will have time to work in small groups on certain problems. Reasonable progress on these will merit full participation points. Particularly nice solutions may merit extra credit.

**Grading:** There will be two grading schemes: One that involves class participation, and one that does not. A decision must be made as to which scheme you prefer within the first two weeks of class.

- Grading Scheme 1
- Final: 30%
- Midterms: 25% (each)
- Homework: 20%
- Grading Scheme 2
- Final: 20%
- Midterms: 15% (each)
- Homework: 20%
- Participation: 30%

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

January 20 | Cartesian, polar, cylindrical and spherical coordinate systems | HW0: 10.3 #(7-12,54); 15.8 #(8,12,13); 15.9#(14,15) | ||

Jan 22 | (15.1) Review of Riemann integral; double integrals | HW1: 15.1 #2,8,10,12,14 | ||

Jan 27 | (15.2-3) Integrals over rectangles and more general regions | HW2A: 15.2 #2,12,20,24,(40) 15.3#8,11,12,14,16,38,(56,62) | ||

Jan 29 | (15.4) Double integrals in polar coordinates | HW2B: 15.4 #1-4,6,10,14,(39,40) | ||

Feb 3 | (15.5-6) Applications; Triple Integrals | HW3A: 15.5 #2,8,28,(32); 15.6#4,10,(22,24) | ||

Feb 5 | (15.7-8) Cylindrical and spherical coordinates | HW3B: 15.7#28,30,34,(48,55a) 15.8#14,(16),30 | ||

Feb 10 | (15.9) Spherical coordinates in Triple integrals | HW4A: 15.9 #18,20,28,(47) | ||

Feb 12 | (15.10) Change of coordinates | HW4B: 15.10 #8,14,18,(24) | ||

Feb 17,19 | Review and catch up; Midterm 1 | |||

Feb 24 | (16.1) Vector Fields | HW5A: 16.1 #4,6,15-18,26,29-32 | ||

Feb 26 | (16.2) Line Integrals | HW5B: 16.2 #6,10, 14,18,34,40,(45,49,50) | ||

Mar 3 | (16.3) The fundamental theorem of line integrals | HW6A: 16.3 #11,14,24,26,28,(29,35) | ||

Mar 5 | (16.4) Green's Theorem | HW6B: 16.4 #2,8,18,19,(21,22),28 | ||

Mar 10 | (16.5-6) Curl and divergence | HW7A: 16.5#9-11,12,16,20,(30,32,37,39); 16.6#4,6,(7-12,13-18),20,(22),24,(30) | ||

Mar 12 | (16.6-7) Surface integrals | HW7B: 16.6#42,46,(64); 16.7#20,39,40 | ||

Mar 24 | (16.8) Stoke's theorem | HW8A: 16.7#24,30 16.8#10,14,16,17 | ||

Mar 26 | (16.9) Divergence theorem | HW8B: 16.9#2,4,12,18,(19),20,24 | ||

Mar 31, Apr 2 | Review and catch up; Midterm 2 | |||

Apr 7,9 | Complex Functions 1 | HW9: The problems from Complex Numbers notes. (Due 4/14.) | ||

Apr 14,16 | Complex Functions 2: Cauchy-Riemann equations | HW10: Problems from Complex Functions and the Cauchy Riemann Equations plus extra problems. (Due 4/21) | ||

Apr 21,23 | Complex Functions 3: Complex line integrals | HW11: Problems from Contour Integrals and Cauchy's Theorem. (Due 4/28) |

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