## What is Calculus IV?

In your wild youth, you studied differentiation (rates of change) and integration (accumulation) and their relationship (the Fundamental Theorem of Calculus). As you mellowed, you learned about multivariable functions. Now you're getting downright old, and we'll talk about vector functions. By the end of this class, as we study Stokes' and Green's Theorems, we'll have enough experience to put everything together into one big picture. We might even address that pesky question: what is it good for?

## Course Syllabus

### Topics

Vector valued functions of one variable (Chapter 14): Parameterized curves, velocity, acceleration, arc length. Includes curvature, normal and binormal vectors, tangential and normal components of acceleration.

Vector valued functions of several variables (Chapter 17): vector fields, line integrals, conservative fields, fundamental theorem of line integrals, Green's theorem, gradient, curl, divergence, parameterized surfaces, suface area, surface integrals, Stokes' theorem, divergence theorem.

### Course Schedule

Wednesday, September 8th, first class

Monday, October 11th, no class

Friday, October 8th, MIDTERM I

Friday, November 5th, MIDTERM II (DATE CHANGED!!)

Friday, December 3rd, last class

TBA, FINAL EXAM

### Textbook

Multivariable Calculus 6th Ed by James Stewart (ISBN-13: 978-0495011637). You do not need the supplementary materials or the package, and older editions or other 'product lines' (such as 'early transcendentals') will do, and I encourage you to find a solution to this problem which fits your budget, but please read the note about problem numbers below. The textbook has a website with some additional resources grouped by chapter (go to chapter 14 or chapter 17). We will cover Chapter 14 (Vector Functions) and Chapter 17 (Vector Calculus). (Other editions change chapter numbers also.)

Important note about problem numbers: Stewart runs a calculus empire, and he has several lines of calculus textbooks which differ in small points, and the editions are updated every year, partly with the purpose of switching around all the problem numbers. To match the assigned homework problems correctly, you must reference this exact book (not an earlier edition, or the "Early Transcendentals" line or the "Early Vectors" line etc.). To avoid requiring you to purchase the textbook just for this rather silly reason, I will make the problems available on my website, and you can also check against the library copy on reserve number QA 303.2 .S735 2008. No points for solving the wrong homework problems!

### Calculators and software

The use of computer software is encouraged (stand-alone calculators are outdated). The place of such tools is to aid learning, and they cannot to be used to do homework for you, and cannot be used on quizzes and exams. It is appropriate to use them to check homework and graded exam answers in order to see what may have gone wrong. The most powerful such online calculator is Sage (http://www.sagemath.org/). After you sign up for an account (click "try sage online"), you can use the software online to graph functions, solve integrals, etc. The graphing capability in particular is extremely powerful and extremely helpful as you learn to visualise equations. For an example of what Sage does, see the Online Demos. Facility with Sage will not be tested on quizzes and exams, but basic use of Sage or other computer software will be assigned as homework. You can use Maple, Mathematica, Matlab, etc. if you prefer (note, I am not an expert in all these, so be prepared to figure them out yourself when necessary).

It is your responsibility read the course policies very carefully. This is our contract, and by taking this course you accept that contract.

## Course Policies

This is our contract, and by taking this course you agree to this contract. It is your responsibility to read it very carefully.

### Attendance and Participation

Attendance in class is expected. University policy: "Regular attendance is expected of students in all their classes (including lectures, laboratories, tutorials, seminars, etc.). Students who neglect their academic work and assignments may be excluded from final examinations." If you miss class for any reason, it is your responsibility to get notes from a classmate. 'Make-up office hour lectures' will not be given. If you habitually arrive late or leave early from class, it may lower your grade (see Grading section, below). You are not permitted to use laptops in class unless you first convince me you have a compelling reason to do so.

Come to every class. Work at a steady pace throughout the semester. I expect you to read the relevant portions of the textbook before and after the corresponding lecture. I expect you to work the problems gradually as the material is covered, not at the last minute. If you are confused about the material, do not avoid the issue. It's normal to be temporarily bewildered sometimes while learning mathematics. Seek help!

### Homework

Homework will be due on Wednesdays, and will generally cover material presented the previous week. Each week you will either turn in written homework at the beginning of class (no homework will be accepted later during the class period) or take a quiz based on the homework during the first 15 minutes of class. This will be determined by a roll of dice. The quizzes will consist of one, two, or three problems taken directly from the homework.

If you do not hand in homework on time, or miss a quiz, for any reason, you will get no credit. If you arrive late for a quiz, you can write it in whatever time remains. At the end of the term, your two lowest homework/quiz grades will be dropped. There will be no make up quizzes or late homework assignments accepted for any reason (reasonable or unreasonable, with or without a doctor's note).

### Midterms & Exams

There will be two midterms, October 8th and November 12th. The final exam is scheduled by the university (TBA). No calculators will be allowed on any of the exams.

### Missing a Midterm

There are legitimate reasons to miss a midterm exam. Some of these are foreeseable (scheduling conflicts with something of greater priority than your full time job of university student, such as religious observances), and for these a request must be made as soon as the conflict is known, and with at least two week's notice, to avoid a grade of zero (and the request may be declined, so the earlier it is made the better). Some of these are unforeeseable (illnesses and family emergencies), and for these you must bring documentation of the event (a dean's or doctor's note). In particular, a doctor's note is only acceptable if it says "the student was too ill to take an exam" instead of "the student was ill". These must be brought to me as soon as possible, preferably by a friend or family member if you are still in hospital. In all cases of a legitimately missed exam, the make-up exam may be administered orally. These policies are in accordance with UBC policy.

Your final grade is given by whichever is greater of the following two computations:

(Homework/Quizzes) x 20% + (First Midterm) x 15% + (Second Midterm) x 15% + (Final Exam) x 50%

OR

(Final Exam) - 10

The second option is your safety net: even if you perform very badly on the midterms, you can still get a good grade in the class by doing well on the final. Final grades will then be scaled to be commensurate with historical averages.

I may elect to increase your grade by 1-2% based on attendance, classroom participation, or dramatic improvement during the semester, etc.

Conversely, I reserve the right to lower your grade by up to 5% if you habitually disrupt the learning of other students during lecture (i.e. talking, using your cell phone, arriving late or leaving early, playing online poker during class, etc.). If you must occasionally arrive late or leave early, simply see me before or after class to give an explanation, and it will not count against you in this manner. (Do recall, however, that homework is due at the beginning of class and quizzes are taken at the beginning of class, so by arriving late on a Wednesday your homework grade will suffer. See the homework section above.)

### Serious Illness or Other Long-Term Disruptions to Course Work

If you are unable to keep up with class work for because of serious chronic illness or other unavoidable chronic disruptions, this should be taken up individually with an advisor in your faculty advising office and with the instructor, to plan strategies for catching up effectively or withdrawing in time for deadlines.

## Homework

Homework will be due on Wednesdays, and will generally cover material presented the previous week. Each week you will either turn in written homework at the beginning of class (no homework will be accepted later during the class period) or take a quiz based on the homework during the first 15 minutes of class. The quizzes will consist of one, two, or three problems taken directly from the homework.

Whether to have a quiz or hand in homework will be decided in class each week on the due date randomly by roll of dice. There are two six-sided dice, one is "homework" and one is "quiz". Both are rolled and the higher one wins. In case of a tie, a coin toss (or other random event with two outcomes of equal probability) will determine whether the class gets "both" or "neither".

If you will miss homework/quizzes, see the course policies.

### How to do your homework for full credit

Do your homework as it is assigned, and not at the last minute. You are only done with a problem when you understand why the methods you used have worked. If all you are doing is blindly applying formulas and mimicking examples, get extra help. The assigned problems are those you hand in; you are expected to do additional problems on your own until you understand the material. Read the Homework Strategies very carefully before you begin your homework.

Prepare your homework according to the following rules for full credit (credit will be deducted for failure to follow these rules):

• Write your name clearly at the top of every page.
• Put the problems in order, indicating clearly what you have skipped.
• Staple your homework. Paper clips, folded corners, etc. are not acceptable.
• Write clearly and indicate why/what you do at each step.

### Assigned homework

Homework #1, Due Wednesday, September 15th:

• Read the course webpage in full. (Nothing to hand in.)
• 14.1 #4, 8, 10, 14, 16, 19-24, 26, 36, 42. See text or Scan 1, Scan 2 for problems.
• SOLUTIONS: here.

Special Homework, Due Monday, September 20th:

• Read Lockhart's Lament for in-class discussion. Many people would claim that reading this one document is more important than taking this class! After reading this, particularly the example with a triangle inside a rectangle, ask yourself, why do I care so much about the difference between an antiderivative and an integral, when the fundamental theorem says we have an equation saying they are "the same"?

Homework #2, Due Wednesday, September 22nd:

• 14.2 #4, 6, 12, 14, 20, 26, 32, 34, 38, 44, 48, 49, 50. Scan 1, Scan 2.
• Either make an account online for SAGE or find other software to use (check out the Online Demos section for resources), and graph 14.1 #29-32. Print it out to hand in with the homework.
• SOLUTIONS: here. Notes: 1) I regret including #48, which is a poorly asked question with no clear "correct answer"; and 2) please read my comments on solutions at Virtual Office Hours.

Homework #3, Due Wednesday, September 29th:

• 14.3 #2, 4, 12 (numerical estimate required: see VOH hints), 14, 16 (VOH hints), 18, 22, 24, 26, 36, 37. Scan 1, Scan 2, Scan 3
• Do 14.3, #34-35 on a computer, print out the result, and write a few sentences about why the graphs are reasonable or surprising and explain some of the key features.
• SOLUTIONS: here (point breakdown: 2/2/3/3/3/3/2/2/3/2/2 & 4 for computer graphs = 31 points)

Homework #4, Due Wednesday, October 6th:

• 14.3 #28, 30, 44, 46, 51, 53, 60. Scan 1, Scan 2, Scan 3
• Consider the line segment between the points (1,0) and (0,1). Find a parametrisation of this line segment in terms of the angle of the vector from the origin (i.e. the parametrisation will be from 0 to pi/2).
• 14.4 #8, 16, 20, 22, 26, 30, 36. Scan 1, Scan 2, Scan 3.
• SOLUTIONS: here

Suggested Review Problems for Midterm #1 (not to hand in):

Midterm #1 will cover 14.1, 14.2, 14.3 and 14.4 EXCEPT for Kepler's Laws of Planetary Motion.
• 14 Review Concept Check: #1-8 are good overview questions. Scan
• 14 True-False with Explanation: all are good but especially #7, 8, 10, 12. Scan 1
• 14 Review Exercises: all are good (this is actually a very even review touching each concept in #1-20); don't bother too much with #7; realise that #21, 22 are just examples of more involved questions; see old exams for further examples. Scan 1, Scan 2
• You can find old exams online; see the math department website and at the ams website.
• Here are some old sample midterm problems from previous Math 317.

Homework #5, Due Wednesday, October 13th:

• Consider an asteriod that revolves around the sun in a circular orbit and has a period of revolution T. If the asteriod is instantaneously stopped in its orbit, it will fall towards the sun. Use Kepler's third law to deduce how long it will take for the asteroid to hit the sun. If the earth were stopped in its orbit, approximately how many days would it take before it hit the sun? (Hint: you can do this question easily if instead you regard this asteriod as almost stopped, so that it goes into a highly eccentric elliptical whose major axis is a bit greater than the radius of the original circular orbit.) HERE IS A SOLUTION (written by another instructory)
• You wake up alone on an earth-like planet with no internet. You soon discover that there is no technology at all on this planet, and no people, but it has a sun and moon much like earth's, plenty of eager-to-please clever monkeys and lots of tasty snails. You are carrying: your calculus textbook, a pencil, a ruler, a stopwatch, a protractor, a microscope (all of very high quality), your lucky gnome, and a marshmallow. You soon realise that the most exciting thing you can do is estimate the mass of the planet. Explain how to do this, using the technology available to you (sticks, stones, birch bark, ...). (Note: if you get stuck and do some research into this question, that's ok, but please cite any resources used.)
• Enjoy Thanksgiving.

MIDTERM #1 SOLUTIONS:

• Midterm Solutions for Test A & Test B (look at the header of your test to figure out which is the right one for you)

Homework #6, Due Wednesday, October 20th:

• 17.1 #2, 6, 8, 11-14, 15-18, 22, 26, 29-32, 34, 36 (maybe try #35 first). Scan (from Early Transcendentals 16.1, identical to our 17.1)
• SOLUTIONS: here

Homework #7, Due Wednesday, October 27th:

• 17.2 #4, 6, 14, 17, 18, 22, 34 (see Example 3 and the discussion before it), 40, 43, 44, 45. Scan
• Do these Extra Problems. These are extremely important extra problems! Do not skip them!
• 17.3 #1, 2, 26. Scan
• SOLUTIONS: book problems, extra problems

Homework #8, Due Wednesday, November 3rd:

• 17.3 #3, 8, 16, 22, 28, 34 Scan
• Extra Problem #1: Write down an example of a region which is 1) not connected; 2) connected but not simply-connected; 3) simply-connected. For examples of what "writing down a region" means, see but don't hand in 17.3 #29-32.
• Extra Problem #2: One consequence of Kepler's Second Law is that a planet in an elliptical orbit moves faster while it is closer to the sun. Explain why this is a consequence of conservation of energy (qualitatively).
• 17.4 #4, 8, 12, 18, 21 (problem #21 may be working ahead a bit, depending how far we get; read the textbook after example 2 and see equation (5), concerning the formulas for area). Scan
• SOLUTIONS: book problems, extra problems

Suggested Review Problems for Midterm #2 (not to hand in):

Midterm #2 will cover all of 14, 17.1-17.3, and nearly all of 17.4 (everything we manage to do in class before the midterm itself; I'll clarify concepts that are *excluded* closer to the midterm date). The emphasis will be on Chapter 17, but I could always throw a Chapter 14 question at you.
• Chapter 14: Review as for Midterm #1.
• Please grade your own old homework, i.e. compare with solutions and learn from it (especially since not all problems are marked by a TA). Especially read the solutions I've written myself (last midterm, some extra problems).
• Chapter 17 Review Concept Check: #1-7. Scan
• Chapter 17 True-False with Explanation: #4,5,6. Scan
• Chapter 17 Review Exercises: #1a), 2-17, 21 (I like - find the simple solution), 23 Scan
• You can find old exams online; see the math department website and at the ams website.
• Here is a packet of review problems.

Homework #9, Due Wednesday, November 10th:

So, I hear you had a midterm recently. But I also hear you already know plenty about the curl and grad (gradient), and are good at reading your textbook. So you shouldn't have any trouble with this assignment. (On Monday we'll talk about the meaning of "divergence," but if you read ahead you'll have no trouble doing these problems before then.)
• 17.5 #4,8,9-11,12,18,20,22. Scan
• SOLUTIONS: HERE.

MIDTERM #2 SOLUTIONS:

Homework #10, Due Wednesday, November 17th:

• It is not to hand in, but look at Nykamp's Readings on Vector Fields, where he discusses the subtleties of divergence and curl very nicely. All of the parts of Section #4a) are relevant reading for this week.
• 17.5 #23-28. You are responsible for knowing and being able to derive identities #23-29, which are "differentiation rules" for div and curl. Here is a worked example of #29; the rest are your homework. Scan
• 17.5 #32 (please look at 30,31 when considering 32, then compare your result with Nykamp's discussion here.) Scan
• 17.5 #33, 34 (these use 23-29; in 33, the book's notation f(grad g) means f times grad g, not f evaluated at grad g!) Scan
• 17.5 #39 (this is more fun if you don't read the hint, which basically gives it all away). Scan
• 17.6 #3, 6 (Don't just "identify" but actually graph these by hand. They're not too bad.) Scan
• 17.6 #8 (you can use the 3D Calc Plotter ("really nice grapher") if you like). Scan
• 17.6 #13-18, 19, 22, 24, 26, 30, 34. Scan
• Extra Problem 1. (a) How would the surface in 17.6 #6 change if the "2t" where changed to "t" in both places it appears? (verify using the grapher if you like) (b) Give a different parametrisation of the surface given in 17.6 #6 which does not involve trigonometric functions. The grid curves will be different, but the surface will be the same surface. Hint: remember the graphing handout we did in class. (c) What is the (old-fashioned, polynomial) equation of this surface in 3 variables (x,y,z)? (Hint: verify your guess by plugging in both parametrisations to see if they satisfy the equation.)
• Extra Problem 2 (for reviewing spherical coordinates). Assume the earth's centre is the origin of a 3D coordinate system. The z-axis points up through the geographic north pole, and the x-axis points out through the Greenwich meridian. Find the (x,y,z) coordinates of you.
• SOLUTIONS: book problems, extra problems

Homework #11, Due Wednesday, November 24th:

• 17.6 #38, 42, 46, 58. Scan
• 17.7 #6, 10, 14, 18, 38. Scan
• Extra Problem 1: Find the average latitude of all points in the northern hemisphere.
• Numerical Answers are up so you can check against them before handing in your solutions.
• SOLUTIONS: book problems, extra problems

Homework #12, Due Wednesday, December 1st:

• 17.7 #20, 22, 28, 42, 44. Scan
• 17.8 #2 For full credit, you must use Stokes' theorem, i.e. do it by a line integral. Scan
• 17.8 #6 For full credit, you must use equation 3, i.e. do it by taking a surface integral of a different surface, no credit for using a line integral on this one. Scan
• 17.8 #8 For full credit, you must use Stokes' theorem, i.e. do it by a surface integral. Scan
• 17.8 #16 The question says 'region enclosed by C' and they mean 'within the plane x+y+z=1'. Scan
• 17.8 #20. Scan
• Check your email concerning a bonus point you can get by filling out an online form before the end of the day Wed (or email me if you haven't gotten that email).
• SOLUTIONS: book problems.

Homework #13, not to hand in:

Although it is important to do this homework, and solutions will be posted, it won't be collected for a grade.
• 17.9 #2, 10, 12, 18 Scan
• 17.9 #20 The textbook finally gets around to talking about what divergence is here; so this is a concept question on divergence. Better late than never. By the way, "sink" means "negative divergence" and "source" means "positive divergence" Scan
• 17.9 #24 Note/hint: This is a *scalar* surface integral. How do we relate it to the divergence theorem? Scan
• 17.9 #28,30 Scan
• SOLUTIONS: USE THESE ONLY *AFTER* doing the problems! Book problems.

Review Problems for Final Exam:

The final exam will cover all of Chapters 14 and 17 of Stewart and everything we have done in lecture and in homework & review problems. A small emphasis will be on Chapter 17.5-9, but it will be cumulative; material from the entire course will be tested.
• Chapter 14: Review as for Midterm #1.
• Chapter 17.1-4: Review as for Midterm #2.
• Please grade your own old homework, i.e. compare with solutions and learn from it (especially since not all problems are marked by a TA). Especially read the solutions I've written myself (midterms, some extra problems). I will test whether you've learned from this stuff.
• Chapter 17 Review Concept Check: all (#8-16 are new). Scan
• Chapter 17 True-False with Explanation: all (#1-3,7-8 are new). Scan
• Chapter 17 Review Exercises: all; the new ones are #1b), 18-19, 20 (they mean, give the conditions as well), 22, 24 (do it the easy way!), 25-41 (always stop and think; what's the easy way!?) Scan
• You can find old exams online; see the math department website and at the ams website.
• The Math Club has printed past exam packages and they are on sale starting next week from Monday to Friday, 11am - 3pm at the Math Club in MATX 1119. The exam packages are \$5 for members and \$10 for non members.
• Review Questions Game Show -- 13 multiple choice: Questions formatted for computer viewing; Questions formatted for printing; Answers.
• Review Packet & Advice. Answers to old exams are available from the UBC Math Club (at certain times of day).

## Homework Strategies

Doing mathematics homework involves a set of specific skills. The first skill is devotion. Since homework is the single best way to cement your understanding of the material in the course, I expect you to do it diligently and do it all. For my part, I will work hard to pick worthwhile problems instead of overwhelming you with volume. Below are some notes and strategies.

2. Before starting your homework, review the section of the textbook we've been covering, make sure your trusty Definitions and Theorems are laid out clearly, and take a glance back at the examples we've done.
3. When reading a problem, make a mental (or written) list of the terms, ideas and strategies that might be relevant. Write down clearly what you need to know and what you already have. Think of ways you've bridged similar gaps in the past, even if the problem is phrased in a novel way. Most of the tools used to bridge such a gap are Theorems we've covered recently.
4. If you are having trouble with one problem, find similar but easier problems in the problem section of the appropriate chapter, and do a few. Choose ones with the answer in the back, and once you feel confident, go back to the one that was giving you trouble. Often you will find you have new understanding.
5. Don't erase when you get stuck; you may want it later. And as soon as you successfully solve a problem, neatly write up your final copy right away, before you lose track of it. If you don't, you'll go back later and suddenly realise what itβs like to be a grader faced with a messy solution.
6. Doing homework with friends can be helpful, since explaining solutions is good practice. When explaining, try to lead by the Socratic method. However, if you find that you receive more explanations than you give, work by yourself before meeting. Remember that homework is preparing you for tests, which you must do alone. And of course, always write up the solution by yourself and in your own words.
7. If you are having trouble solving problems, perhaps you should work on problem solving strategies. Ask a tutor, or in office hours, for specific help with this.
8. Working on your homework sets at the Drop-In Tutoring Centre can be a good way of getting extra help. Take advantage of this and other resources listed under Getting Help.
9. STAPLE YOUR HOMEWORK. You can buy a lifetime supply of staples at β where else? β Staples for under a dollar. Yes, the lifetime friendship of your grader is that cheap. Please also write your name clearly at the top, and keep the problems in the correct order (indicating any that you skipped).

## Getting Help

Being confused is part of mathematics. In fact, as a research mathematician, that's basically my job description: to be confused about things all day long. So I'm no foreigner to the frustrations of learning new mathematics. The key is to use all your resources to tackle that confusion. A big part of that is learning how to learn math: how to study, how to do homework, and how to help yourself when you get stuck. Below are some tips on how to succeed.

### General Study Suggestions

Learning mathematics is like learning a language. There are three golden rules:

1) Spend a little time every workday. You cannot learn Russian the day before your Russian exam by reading the dictionary. You will do much better by putting aside an hour a day to work on your homework than by leaving it for the night before. One tip: count your time in hours, not completed homework problems. Relax and enjoy your date with math: explore ideas and don't rush.

2) Practice! To learn Russian, you need to speak it. Similarly, you need to learn mathematics by doing practice problems. If you are reading the textbook and it looks like nonsense, choose a simple example and try to work through it yourself. You will learn much more than the textbook could ever tell you. And, just like with a language, practicing with friends and mentors is even better.

3) Make mistakes! (Do it now so you don't have to do it on the final exam.) To learn to speak Russian, you have to dive in and try! And when you first start speaking, you won't get a single tense right. If you are paralysed by the need to do a problem correctly the first time, you will never solve it. You must try things and see why they do or do not work. Every mathematician and non-mathematician does this. My job description as a graduate student is something like this: "Bangs head against wall for extended periods and writes down many wrong things. Eventually writes down something correct once every month or so." So don't be afraid: I spend more of my day making mistakes than you do. I know what it's like.

Here are some ways to practice making mistakes:

• try something you don't think will work and see if it does
• write in pen and cross out errors gently -- you may find out your previous work wasn't as useless as you first thought
• try to do a problem more than one way and see if you get the same answer

And finally...

Remember, no one understands everything the first time. Learning math means going over things in lecture, then at home, then with the textbook, then with your homework, then in section, then in office hours, then in review sessions...

Still afraid of mathematics? This really does happen, even to math majors! If you want to talk about this, feel free to come by.

### Virtual Office Hours

Emailed student questions will be posted (anonymously & possibly edited) to Virtual Office Hours, where they will be answered. Please check it frequently, especially for notes on this week's homework.

### Office Hours

My office hours are Mondays 12-2 and Fridays 11-12 in Auditorium Annex A, room 149. This is shared office space, so we cannot start early or continue late. I have a joint appointment with SFU, so I am available for appointments on UBC campus on MWF only. Extra office hour appointments will not be private unless specifically requested (I may double-schedule).

### Tutorial Centre

UBC runs a tutorial centre for calculus students. Click here for the schedule.

### AMS Tutoring

Even if you aren't in first or second year, the tutoring program run by AMS can provide connections to tutors.

### Textbook on reserve

The I.K. Barber Learning Centre has a copy on reserve: QA 303.2 .S735 2008.

### Department Resources

The math department has a variety of undergraduate resources.

### Past Exams

Past exams can be found at the math department website and at the ams website.

## Online Resources and Computer Demos (including SAGE)

Updated frequently.